I: 02, 18-33, LNM 39 (1967)
CAIROLI, Renzo
Semi-groupes de transition et fonctions excessives (
Markov processes,
Potential theory)
A study of product kernels, product semi-groups and product Markov processes
Comment: This paper was the first step in R.~Cairoli's study of two-parameter processes
Keywords: Product semigroupsNature: Original
Retrieve article from Numdam
I: 04, 52-53, LNM 39 (1967)
DELLACHERIE, Claude
Un complément au théorème de Weierstrass-Stone (
Functional analysis)
An easy but useful remark on the relation between the ``lattice'' and ``algebra'' forms of Stone's theorem, which apparently belongs to the folklore
Keywords: Stone-Weierstrass theoremNature: Original Retrieve article from Numdam
I: 05, 54-71, LNM 39 (1967)
FERNIQUE, Xavier
Séries de distributions aléatoires indépendantes (2 talks) (
Miscellanea)
This is part of X.~Fernique's research on random distributions (probability measures on ${\cal D}'$, and more generally on the dual space $E'$ of a nuclear LF space $E$) and their characteristic functions, which are exactly, according to Minlos' theorem, the continuous positive definite functions on $E$ assuming the value $1$ at $0$. Here it is proved that a series of independent random distributions converges a.s. if and only if the product of their characteristic functions converges pointwise to a continuous limit, and converges a.s. after centering if and only if the product of absolute values converges
Comment: See for further results
Ann. Inst. Fourier, 17-1, 1967;
Invent. Math.,
3, 1967, and
C.R. Acad. Sc.,
266, 1968 for the extension of Lévy's continuity theorem (also presented at Séminaire Bourbaki, June 1966,
311)
Keywords: Random distributions,
Minlos theoremNature: Original Retrieve article from Numdam
I: 06, 72-162, LNM 39 (1967)
MEYER, Paul-André
Intégrales stochastiques I--IV (4 talks) (
Martingale theory,
Stochastic calculus)
This series presents an expanded exposition of the celebrated paper of Kunita-Watanabe (
Nagoya Math. J. 30, 1967) on square integrable martingales. The filtration is assumed to be free from fixed times of discontinuity, a restriction lifted in the modern theory. A new feature is the definition of the second increasing process associated with a square integrable martingale (a ``square bracket'' in the modern terminology). In the second talk, stochastic integrals are defined with respect to local martingales (introduced from Ito-Watanabe,
Ann. Inst. Fourier, 15, 1965), and the general integration by parts formula is proved. Also a restricted class of semimartingales is defined and an ``Ito formula'' for change of variables is given, different from that of Kunita-Watanabe. The third talk contains the famous Kunita-Watanabe theorem giving the structure of martingale additive functionals of a Hunt process, and a new proof of Lévy's description of the structure of processes with independent increments (in the time homogeneous case). The fourth talk deals mostly with Lévy systems (Motoo-Watanabe,
J. Math. Kyoto Univ.,
4, 1965; Watanabe,
Japanese J. Math.,
36, 1964)
Comment: This paper was a step in the development of stochastic integration. Practically every detail of it has been reworked since, starting with Doléans-Dade-Meyer
409. Note a few corrections in Meyer
312Keywords: Square integrable martingales,
Angle bracket,
Stochastic integralsNature: Exposition,
Original additions Retrieve article from Numdam
I: 07, 163-165, LNM 39 (1967)
MEYER, Paul-André
Sur un théorème de Deny (
Potential theory,
Measure theory)
In the potential theory of a resolvent which satisfies the absolute continuity hypothesis, every sequence of excessive functions contains a subsequence which converges except on a set of potential zero. It is also proved that a sequence which converges weakly in $L^1$ but not strongly must oscillate around its limit
Comment: a version of this result in classical potential theory was proved by Deny,
C.R. Acad. Sci.,
218, 1944. The cone of excessive functions possesses good compactness properties, discovered by Mokobodzki. See Dellacherie-Meyer,
Probabilités et Potentiel, end of chapter XII
Keywords: A.e. convergence,
SubsequencesNature: Original Retrieve article from Numdam
II: 01, 1-21, LNM 51 (1968)
AZÉMA, Jacques;
DUFLO, Marie;
REVUZ, Daniel
Classes récurrentes d'un processus de Markov (
Markov processes)
This is an improved version of a paper by the same authors (
Ann. Inst. H. Poincaré, 2, 1966). Its aim is a theory of recurrence in continuous time (for a Hunt process). The main point is to use the finely open sets instead of the ordinary ones to define recurrence
Comment: The subject is further investigated by the same authors in
302Keywords: Recurrent sets,
Fine topologyNature: Original Retrieve article from Numdam
II: 02, 22-33, LNM 51 (1968)
CARTIER, Pierre;
MEYER, Paul-André;
WEIL, Michel
Le retournement du temps~: compléments à l'exposé de M.~Weil (
Markov processes)
In
108, M.~Weil had presented the work of Nagasawa on the time reversal of a Markov process at a ``L-time'' or return time. Here the results are improved on three points: a Markovian filtration is given for the reversed process; an analytic condition on the semigroup is lifted; finally, the behaviour of the
coexcessive functions on the sample functions of the original process is investigated
Comment: The results of this paper have become part of the standard theory of time reversal. See
312 for a correction
Keywords: Time reversal,
Dual semigroupsNature: Original Retrieve article from Numdam
II: 03, 34-42, LNM 51 (1968)
DOLÉANS-DADE, Catherine
Fonctionnelles additives parfaites (
Markov processes)
The identity defining additive (or multiplicative) functionals involves an exceptional set depending on a continuous time $t$. If the exceptional set can be chosen independently of $t$, the functional is perfect. It is shown that every additive functional of a Hunt process admitting a reference measure has a perfect version
Comment: The existence of a reference measure was lifted by Dellacherie in
304. However, the whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see
623Keywords: Additive functionals,
PerfectionNature: Original Retrieve article from Numdam
II: 09, 166-170, LNM 51 (1968)
MEYER, Paul-André
Une majoration du processus croissant associé à une surmartingale (
Martingale theory)
Let $(X_t)$ be the potential generated by a previsible increasing process $(A_t)$. Then a norm equivalence in $L^p,\ 1<p<\infty$ is given between the random variables $X^\ast$ and $A_\infty$
Comment: This paper became obsolete after the $H^1$-$BMO$ theory
Keywords: Inequalities,
Potential of an increasing processNature: Original Retrieve article from Numdam
II: 11, 175-199, LNM 51 (1968)
MEYER, Paul-André
Compactifications associées à une résolvante (
Potential theory)
Let $E$ be a locally compact space, $(U_p)$ be a submarkovian resolvent, with a potential kernel $U=U_0$ which maps $C_k$ (the continuous functions with compact support) into continuous bounded functions. Let $F$ be a compact space containing $E$ as a dense subset, but inducing possibly a coarser topology. It is assumed that all potentials $Uf$ with $f\in C_k$ extend to continuous functions on $F$, and that points of $F$ are separated by continuous functions on $F$ whose restriction to $E$ is supermedian. Then it is shown how to extend the resolvent to $F$ and imitate the construction of a Ray semigroup and a strong Markov process. This was an attempt to compactify the space using only supermedian functions, not $p$-supermedian for all $p>0$. An application to Markov chains is given
Comment: This method of compactification suggested by Chung's boundary theory for Markov chains (similarly Doob,
Trans. Amer. Math. Soc.,
149, 1970) never superseded the standard Ray-Knight approach
Keywords: Resolvents,
Ray compactification,
Martin boundary,
Boundary theoryNature: Original Retrieve article from Numdam
III: 02, 24-33, LNM 88 (1969)
AZÉMA, Jacques;
DUFLO, Marie;
REVUZ, Daniel
Mesure invariante des processus de Markov récurrents (
Markov processes)
A condition similar to the Harris recurrence condition is studied in continuous time. It is shown that it implies the existence (up to a constant factor) of a unique $\sigma$-finite excessive measure, which is invariant. The invariant measure for a time-changed process is described
Comment: This is related to several papers by the same authors on recurrent Markov processes, and in particular to
201Keywords: Recurrent potential theory,
Invariant measuresNature: Original Retrieve article from Numdam
III: 03, 34-92, LNM 88 (1969)
CAIROLI, Renzo
Étude probabiliste d'un problème de Dirichlet (
Several parameter processes)
This paper aims at a better understanding of separately harmonic functions with respect to a product of two Markov processes, which provide one of the main examples of two parameter martingales (the other one being the Brownian sheet). Here the recently published work of J.~Walsh (
Ann. Inst. Fourier, 18, 1968) on the Dirichlet problem for biharmonic functions was discussed and reinterpreted
Comment: An early step in the theory of two parameter martingales. See also Cairoli,
Publ. Inst. Stat. Univ. Paris, 15, 1966
Keywords: Dirichlet problem,
Biharmonic functionsNature: Original Retrieve article from Numdam
III: 04, 93-96, LNM 88 (1969)
DELLACHERIE, Claude
Une application aux fonctionnelles additives d'un théorème de Mokobodzki (
Markov processes)
Mokobodzki showed the existence of ``rapid ultrafilters'' on the integers, with the property that applied to a sequence that converges in probability they converge a.s. (see for instance Dellacherie-Meyer,
Probabilité et potentiels, Chap. II,
27). They are used here to prove that every continuous additive functional of a Markov process has a ``perfect'' version
Comment: See also
203. The whole subject of perfect additive functionals has been closed by Walsh's approach using the essential topology, see
623Keywords: Additive functionals,
PerfectionNature: Original Retrieve article from Numdam
III: 05, 97-114, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires I (
Descriptive set theory,
Markov processes,
General theory of processes)
A deep theorem of Lusin asserts that a Borel set with countable sections is a countable union of Borel graphs. It is applied here in the general theory of processes to show that an optional set with countable sections is a countable union of graphs of stopping times, and in the theory of Markov processes, that a Borel set which is a.s. hit by the process at countably many times must be semi-polar
Comment: See Dellacherie,
Capacités et Processus Stochastiques, Springer 1972
Keywords: Sets with countable sectionsNature: Original Retrieve article from Numdam
III: 06, 115-136, LNM 88 (1969)
DELLACHERIE, Claude
Ensembles aléatoires II (
Descriptive set theory,
Markov processes)
Among the many proofs that an uncountable Borel set of the line contains a perfect set, a proof of Sierpinski (
Fund. Math.,
5, 1924) can be extended to an abstract set-up to show that a non-semi-polar Borel set contains a non-semi-polar compact set
Comment: See Dellacherie,
Capacités et Processus Stochastiques, Springer 1972. More recent proofs no longer depend on ``rabotages'': Dellacherie-Meyer,
Probabilités et potentiel, Appendix to Chapter IV
Keywords: Sierpinski's ``rabotages'',
Semi-polar setsNature: Original Retrieve article from Numdam
III: 07, 137-142, LNM 88 (1969)
HUBER, Catherine
Un aspect de la loi du logarithme itéré pour des variables aléatoires indépendantes et équidistribuées (
Independence)
A refinement of the classical law of the iterated logarithm for i.i.d. random variables, under regularity assumptions on the tail of the common distribution
Keywords: Law of the iterated logarithmNature: Original Retrieve article from Numdam
III: 09, 144-151, LNM 88 (1969)
MEYER, Paul-André
Un résultat de théorie du potentiel (
Potential theory,
Markov processes)
Under strong duality hypotheses, it is shown that a measure which does not charge polar sets is equivalent to a measure whose Green potential is bounded
Comment: See Meyer,
Processus de Markov, Lecture Notes in M.
26Keywords: Green potentials,
Dual semigroupsNature: Original Retrieve article from Numdam
III: 10, 152-154, LNM 88 (1969)
MEYER, Paul-André
Un résultat élémentaire sur les temps d'arrêt (
General theory of processes)
This useful result asserts that a stopping time is accessible if and only if its graph is contained in a countable union of graphs of previsible stopping times
Comment: Before this was noticed, accessible stopping times were considered important. After this remark, previsible stopping times came to the forefront
Keywords: Stopping times,
Accessible times,
Previsible timesNature: Original Retrieve article from Numdam
III: 11, 155-159, LNM 88 (1969)
MEYER, Paul-André
Une nouvelle démonstration des théorèmes de section (
General theory of processes)
The proof of the section theorems has improved over the years, from complicated-false to complicated-true, and finally to easy-true. This was a step on the way, due to Dellacherie (inspired by Cornea-Licea,
Z. für W-theorie, 10, 1968)
Comment: This is essentially the definitive proof, using a general section theorem instead of capacity theory
Keywords: Section theorems,
Optional processes,
Previsible processesNature: Original Retrieve article from Numdam
III: 15, 190-229, LNM 88 (1969)
MORANDO, Philippe
Mesures aléatoires (
Independent increments)
This paper consists of two talks, on the construction and structure of measures with independent values on an abstract measurable space, inspired by papers of Prekopa (
Acta Math. Acad. Sci. Hung., 7, 1956 and
8, 1957) and Kingman (
Pacific J. Math., 21, 1967)
Comment: If the measurable space is not ``too'' abstract, it can be imbedded into the line, and the standard theory of Lévy processes (non-homogeneous) can be used. This simple remark reduces the interest of the general treatment: see Dellacherie-Meyer,
Probabilités et potentiel, Chapter XIII, end of \S4
Keywords: Random measures,
Independent incrementsNature: Exposition,
Original additions Retrieve article from Numdam
IV: 01, 1-27, LNM 124 (1970)
CAIROLI, Renzo
Une inégalité pour martingales à indices multiples et ses applications (
Several parameter processes)
This paper was the starting point of the theory of two-parameter martingales. It proves the corresponding Doob inequality and convergence theorem, with an application to biharmonic functions
Comment: The next landmark in the theory is Cairoli-Walsh,
Acta. Math.,
134, 1975. For the modern results, see Imkeller,
Two Parameter Processes and their Quadratic Variation, Lect. Notes in M.
1308, 1989
Keywords: Two-parameter martingales,
Maximal inequality,
Almost sure convergenceNature: Original Retrieve article from Numdam
IV: 03, 37-46, LNM 124 (1970)
CHERSI, Franco
Martingales et intégrabilité de $X\log^+X$ d'après Gundy (
Martingale theory)
Gundy's result (
Studia Math.,
33, 1968) is a converse to Doob's inequality: for a positive martingale such that $X_n\leq cX_{n-1}$, the integrability of $\sup_n X_n$ implies boundedness in $L\log^+L$. All martingales satisfy this condition on regular filtrations
Comment: The integrability of $\sup_n |\,X_n\,|$ has become now the $H^1$ theory of martingales
Keywords: Inequalities,
Regular martingalesNature: Exposition,
Original additions Retrieve article from Numdam
IV: 05, 60-70, LNM 124 (1970)
DELLACHERIE, Claude
Un exemple de la théorie générale des processus (
General theory of processes)
In the case of the smallest filtration for which a given random variable is a stopping time, all the computations of the general theory can be performed explicitly
Comment: This example has become classical. See for example Dellacherie-Meyer,
Probabilités et Potentiel, Chap IV. On the other hand, it can be extended to deal with (unmarked) point processes: see Chou-Meyer
906Keywords: Stopping times,
Accessible times,
Previsible timesNature: Original Retrieve article from Numdam
IV: 06, 71-72, LNM 124 (1970)
DELLACHERIE, Claude
Au sujet des sauts d'un processus de Hunt (
Markov processes)
Two a.s. results on jumps: the process cannot jump
from a semi-polar set; at the first hitting time of any finely closed set $F$, either the process does not jump, or it jumps from outside $F$
Comment: Both results are improvements of previous results of Meyer and Weil
Keywords: Hunt processes,
Semi-polar setsNature: Original Retrieve article from Numdam
IV: 07, 73-75, LNM 124 (1970)
DELLACHERIE, Claude
Potentiels de Green et fonctionnelles additives (
Markov processes,
Potential theory)
Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$
Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces
Keywords: Green potentials,
Additive functionalsNature: Original Retrieve article from Numdam
IV: 08, 76-76, LNM 124 (1970)
DELLACHERIE, Claude
Un lemme de théorie de la mesure (
Measure theory)
A lemma used by Erdös, Kesterman and Rogers (
Coll. Math., XI, 1963) is reduced to the fact that a sequence of bounded r.v.'s contains a weakly convergent subsequence
Keywords: Convergence in norm,
SubsequencesNature: Original proofs Retrieve article from Numdam
IV: 09, 77-107, LNM 124 (1970)
DOLÉANS-DADE, Catherine;
MEYER, Paul-André
Intégrales stochastiques par rapport aux martingales locales (
Martingale theory,
Stochastic calculus)
This is a continuation of Meyer
106, with a new complete exposition of the theory, and two substantial improvements: the filtration is general (while in
106 it was assumed free of fixed times of discontinuity) and the definition of semimartingales is the modern one (while in
106 they were the special semimartingales of nowadays). The change of variables formula is given in its full generality
Comment: The results of this paper have become classical, and are reproduced almost literally in Meyer
1017Keywords: Local martingales,
Stochastic integrals,
Change of variable formulaNature: Original Retrieve article from Numdam
IV: 10, 108-131, LNM 124 (1970)
FUCHS, Aimé;
LETTA, Giorgio
L'inégalité de Kullback. Application à la théorie de l'estimation (
Information theory,
Mathematical statistics)
This paper brings together, and shows the mutual relation of a number of classical definitions, centering on Shannon's mutual information of two probability measures, $G(\lambda|\mu)=\int \log {d\lambda\over d\mu}\,d\mu$
Comment: To be asked from the authors
Keywords: Kullback inequality,
Cramer-Rao inequality,
Sufficient statisticsNature: Original Retrieve article from Numdam
IV: 11, 132-132, LNM 124 (1970)
SAM LAZARO, José de
Théorème de Stone et espérances conditionnelles (
Ergodic theory)
It is shown that the spectral projections of the unitary group arising from a group of measure preserving transformations must be complex operators, and in particular cannot be conditional expectations
Comment: This remark arose from the work on flows in Sam Lazaro-Meyer,
Z. für W-theorie, 18, 1971
Keywords: Flows,
Spectral representationNature: Original Retrieve article from Numdam
IV: 13, 151-161, LNM 124 (1970)
MAISONNEUVE, Bernard;
MORANDO, Philippe
Temps locaux pour les ensembles régénératifs (
Markov processes)
This paper uses the results of the preceding one
412 to define and study the local time of a perfect regenerative set with empty interior (e.g. the set of zeros of Brownian motion), a continuous adapted increasing process whose set of points of increase is exactly the given set
Comment: Same references as the preceding paper
412Keywords: Renewal theory,
Regenerative sets,
Local timesNature: Original Retrieve article from Numdam
IV: 15, 170-194, LNM 124 (1970)
MOKOBODZKI, Gabriel
Densité relative de deux potentiels comparables (
Potential theory)
The main problem considered here is the following: given a transient resolvent $(V_{\lambda})$ on a measurable space, a finite potential $Vg$, an excessive function $u$ dominated by $Vg$ in the strong sense (i.e., $Vg-u$ is excessive), show that $u=Vf$ for some $f\leq g$, and compute $f$ by some ``derivation'' procedure, like $\lim_{\lambda\rightarrow\infty} \lambda(I-\lambda V_{\lambda})\,u$
Comment: The main theorem and the technical tools of its proof have been landmarks in the potential theory of a resolvent, though in the case of the resolvent of a good Markov process there is a simple probabilistic proof of the main result. Another exposition can be found in
Séminaire Bourbaki, 422, November 1972. See also Chapter XII of Dellacherie-Meyer,
Probabilités et potentiel, containing new proofs due to Feyel
Keywords: Resolvents,
Strong ordering,
Lebesgue derivation theoremNature: Original Retrieve article from Numdam
IV: 16, 195-207, LNM 124 (1970)
MOKOBODZKI, Gabriel
Quelques propriétés remarquables des opérateurs presque positifs (
Potential theory)
A sequel to the preceding paper
415. Almost positive operators are candidates to the role of derivation operators relative to a resolvent
Comment: Same as
415Keywords: Resolvents,
Strong ordering,
Lebesgue derivation theoremNature: Original Retrieve article from Numdam
IV: 17, 208-215, LNM 124 (1970)
REVUZ, Daniel
Application d'un théorème de Mokobodzki aux opérateurs potentiels dans le cas récurrent (
Potential theory,
Markov processes)
Mokododzki's theorem asserts that if the kernels of a resolvent are strong Feller, i.e., map bounded functions into continuous functions, then they must satisfy a norm continuity property (see
210). This is used to show the existence for``normal'' recurrent processes of a nice potential operator, defined for suitable functions of zero integral with respect to the invariant measure
Comment: For additional work of Revuz on recurrence, see
Ann. Inst. Fourier, 21, 1971
Keywords: Recurrent potential theoryNature: Original Retrieve article from Numdam
IV: 18, 216-239, LNM 124 (1970)
WEIL, Michel
Quasi-processus (
Markov processes)
Excessive measures which are not potentials of measures were shown by Hunt (
Ill. J. Math.,
4, 1960) to be associated with a probabilistic object which is a kind of projective limit of Markov processes. Hunt's construction was performed in discrete time only, and is difficult in continuous time because of measure theoretic difficulties (the standard theorem on projective limits cannot be applied). Here the construction is done in full detail
Comment: Further work by M.~Weil on the same subject in
532; see the references there
Keywords: Hunt quasi-processesNature: Original Retrieve article from Numdam
V: 01, 1-16, LNM 191 (1971)
ARTZNER, Philippe
Fonctions caractéristiques et mesures planes invariantes par rotation (
Miscellanea)
A study of the class of probability measures on the line which are projections of a measure on the plane invariant by rotation
Keywords: Characteristic functionsNature: Original Retrieve article from Numdam
V: 03, 21-36, LNM 191 (1971)
BRETAGNOLLE, Jean
Résultats de Kesten sur les processus à accroissements indépendants (
Markov processes,
Independent increments)
The question is to find all Lévy processes for which single points are polar. Kesten's answer (
Mem. Amer. Math. Soc.,
93, 1969) is almost complete and in particular proves Chung's conjecture. The proofs in this paper have been considerably reworked
Comment: See also
502 in the same volume
Keywords: Subordinators,
Polar setsNature: Exposition,
Original additions Retrieve article from Numdam
V: 04, 37-57, LNM 191 (1971)
CAIROLI, Renzo
Décomposition de processus à indices doubles (
Several parameter processes)
A discrete submartingale is decomposed into an increasing process and three different kinds of ``martingales''. Extension to continuous time. Earlier than the fundamental paper of Cairoli-Walsh (
Acta Math.,
134, 1975)
Comment: See Cairoli
401Keywords: Two-parameter martingalesNature: Original Retrieve article from Numdam
V: 07, 77-81, LNM 191 (1971)
DELLACHERIE, Claude
Quelques commentaires sur les prolongements de capacités (
Descriptive set theory)
Remarks on the extension of capacities from sets to functions. Probably superseded by the work of Mokobodzki on functional capacities
Comment: See Dellacherie-Meyer,
Probabilités et Potentiel, Chap. XI: capacités fonctionnelles
Keywords: CapacitiesNature: Original Retrieve article from Numdam
V: 08, 82-85, LNM 191 (1971)
DELLACHERIE, Claude
Une démonstration du théorème de séparation des ensembles analytiques (
Descriptive set theory)
The first separation theorem can be deduced from Choquet's capacity theorem
Comment: Starting point in Sion,
Ann. Inst. Fourier, 13, 1963. This proof has become standard, see Dellacherie-Meyer,
Probabilités et Potentiel, Chap. III
Keywords: Analytic sets,
Capacities,
Separation theoremNature: Original Retrieve article from Numdam
V: 09, 86-86, LNM 191 (1971)
DELLACHERIE, Claude
Correction à ``Ensembles Aléatoires II'' (
Descriptive set theory)
Correction to Dellacherie
306Comment: See Dellacherie
511Keywords: Sierpinski's ``rabotages''Nature: Original Retrieve article from Numdam
V: 11, 103-126, LNM 191 (1971)
DELLACHERIE, Claude
Ensembles pavés et rabotages (
Descriptive set theory)
A systematic study of the ``rabotages de Sierpinski'', used in Dellacherie
306 to solve several problems in probabilistic potential theory. The main paper on this subject
Comment: See Dellacherie,
Capacités et Processus Stochastiques, 1970. Author should be consulted on recent developments (see
1526)
Keywords: Analytic sets,
Capacities,
Sierpinski's ``rabotages''Nature: Original Retrieve article from Numdam
V: 12, 127-137, LNM 191 (1971)
DELLACHERIE, Claude;
DOLÉANS-DADE, Catherine
Un contre-exemple au problème des laplaciens approchés (
Martingale theory)
The ``approximate Laplacian'' method of computing the increasing process associated with a supermartingale does not always converge in the strong sense: solves a problem open for many years
Comment: Problem originated in Meyer,
Ill. J. Math.,
7, 1963
Keywords: Submartingales,
SupermartingalesNature: Original Retrieve article from Numdam
V: 13, 138-140, LNM 191 (1971)
DOLÉANS-DADE, Catherine
Une martingale uniformément intégrable, non localement de carré intégrable (
Martingale theory)
Now well known! This paper helped to set the basic notions of the theory
Keywords: Square integrable martingalesNature: Original Retrieve article from Numdam
V: 14, 141-146, LNM 191 (1971)
DOLÉANS-DADE, Catherine
Intégrales stochastiques par rapport à une famille de probabilités (
Stochastic calculus)
Given a family of probability laws on the same space, construct versions of stochastic integrals which do not depend on the law
Comment: Expanded by Stricker-Yor, Calcul stochastique dépendant d'un paramètre,
Z. für W-theorie, 45, 1978
Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
V: 15, 147-169, LNM 191 (1971)
MAISONNEUVE, Bernard
Ensembles régénératifs, temps locaux et subordinateurs (
General theory of processes,
Renewal theory)
New approach to the theory of regenerative sets (Kingman; Krylov-Yushkevic 1965, Hoffmann-Jørgensen,
Math. Scand.,
24, 1969), including a general definition of local time of a random set
Comment: See Meyer
412, Morando-Maisonneuve
413, later work of Maisonneuve in
813 and later
Keywords: Local times,
Subordinators,
Renewal theoryNature: Original Retrieve article from Numdam
V: 17, 177-190, LNM 191 (1971)
MEYER, Paul-André
Processus de Poisson ponctuels d'après K. Ito (
Markov processes,
Point processes)
Presents (a preliminary form of) the celebrated paper of Ito (
Proc. Sixth Berkeley Symposium, 3, 1972) on excursion theory, with an extension (the use of possibly unbounded entrance laws instead of initial measures) which has become part of the now classical theory
Comment: A slip in the definition of Poisson point processes is corrected in vol. VI p.253. The material has appeared repeatedly in book form
Keywords: Poisson point processes,
Excursions,
Local timesNature: Exposition,
Original additions Retrieve article from Numdam
V: 18, 191-195, LNM 191 (1971)
MEYER, Paul-André
Démonstration simplifiée d'un théorème de Knight (
Martingale theory)
A well known theorem (Dambis, Dubins) asserts that a continuous martingale reduces to Brownian motion when time-changed by its own increasing process. Knight's theorem (LN in M
190) asserts that this operation performed on $n$ orthogonal martingales yields $n$ independent Brownian motions. The result is extended to Poisson processes
Comment: Still simpler proofs can be given, see
1448 (included in Revuz-Yor
Continuous Martingales and Brownian Motion, Chapter V)
Keywords: Continuous martingales,
Changes of timeNature: Exposition,
Original additions Retrieve article from Numdam
V: 21, 211-212, LNM 191 (1971)
MEYER, Paul-André
Deux petits résultats de théorie du potentiel (
Potential theory)
Excessive functions are characterized by their domination property over potentials. The strong ordering relation between two functions is carried over to their réduites
Comment: See Dellacherie-Meyer
Probability and Potentials, Chapter XII, \S2
Keywords: Excessive functions,
Réduite,
Strong orderingNature: Original Retrieve article from Numdam
V: 22, 213-236, LNM 191 (1971)
MEYER, Paul-André
Le retournement du temps, d'après Chung et Walsh (
Markov processes)
The paper of Chung and Walsh (
Acta Math.,
134, 1970) proved that any right continuous strong Markov process had a reversed left continuous moderate Markov process at any $L$-time, with a suitably constructed dual semigroup. Appendix 1 gives a useful characterization of càdlàg processes using stopping times (connected with amarts). Appendix 2 proves (following Mokobodzki) that any excessive function strongly dominated by a potential of function is such a potential
Comment: The theorem of Chung-Walsh remains the deepest on time reversal (to be supplemented by the consideration of Kuznetsov's measures)
Keywords: Time reversal,
Dual semigroupsNature: Exposition,
Original additions Retrieve article from Numdam
V: 23, 237-250, LNM 191 (1971)
MEYER, Paul-André
Travaux de H. Rost en théorie du balayage (
Potential theory,
Ergodic theory)
The ``filling scheme'' is a technique used in ergodic theory to prove Hopf's maximal Lemma and the Chacon-Ornstein theorem, studied in detail by H.~Rost (
Zeit. für W-theorie, 15, 1970;
Ann. Inst. Fourier, 21, 1971): it provides a solution to Skorohod's imbedding problem for measures on discrete time Markov processes. Here it is also used to prove Brunel's Lemma in pointwise ergodic theory
Comment: Extension to continuous time in Meyer
612. See also
806,
1012. A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Filling scheme,
Brunel's lemma,
Skorohod imbeddingNature: Exposition,
Original additions Retrieve article from Numdam
V: 24, 251-269, LNM 191 (1971)
MEYER, Paul-André
Solutions de l'équation de Poisson dans le cas récurrent (
Potential theory,
Markov processes)
The problem is to solve the Poisson equation for measures, $\mu-\mu P=\theta$ for given $\theta$, in the case of a recurrent transition kernel $P$. Here a ``filling scheme'' technique is used
Comment: The paper was motivated by Métivier (
Ann. Math. Stat.,
40, 1969) and is completely superseded by one of Revuz (
Ann. Inst. Fourier, 21, 1971)
Keywords: Recurrent potential theory,
Filling scheme,
Harris recurrence,
Poisson equationNature: Original Retrieve article from Numdam
V: 26, 275-277, LNM 191 (1971)
REVUZ, Daniel
Remarque sur les potentiels de mesure (
Markov processes,
Potential theory)
The standard proof of the equivalence between semi-polar sets being polar and a very precise domination principle (Blumenthal-Getoor,
Markov Processes and Potential Theory, 1968) uses the assumption that excessive functions are lower semicontinuous. This assumption is weakened
Comment: To be asked
Keywords: Polar sets,
Semi-polar sets,
Excessive functionsNature: Original Retrieve article from Numdam
V: 27, 278-282, LNM 191 (1971)
SAM LAZARO, José de;
MEYER, Paul-André
Une remarque sur le flot du mouvement brownien (
Brownian motion,
Ergodic theory)
It is proved that the second Wiener chaos (for Brownian motion over the line with its time-invariant measure) contains infinitely many screw-lines orthogonal in the weak sense
Comment: See Sam Lazaro-Meyer,
Z. für W-theorie, 18, 1971
Keywords: Brownian motion,
Wiener chaos,
Screw-linesNature: Original Retrieve article from Numdam
V: 28, 283-289, LNM 191 (1971)
WALSH, John B.
Two footnotes to a theorem of Ray (
Markov processes)
Ray's theorem (
Ann. of Math.,
70, 1959) is the construction of a good semigroup (and process) from a Ray resolvent. The first ``footnote'' gives the construction of a second semigroup with nice properties from the left instead of the right side. The second ``footnote'' studies the filtration of a Ray process
Comment: See Meyer-Walsh,
Invent. Math.,
14, 1971
Keywords: Ray compactificationNature: Original Retrieve article from Numdam
V: 29, 290-310, LNM 191 (1971)
WALSH, John B.
Some topologies connected with Lebesgue measure (
Markov processes,
General theory of processes,
Potential theory)
It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called
essential topology, used in the paper of Chung and Walsh
522 in the same volume
Comment: See Doob
Bull. Amer. Math. Soc.,
72, 1966. An important application in given by Walsh
623 in the next volume. See the paper
1025 of Benveniste. For the use of a different topology see Ito
J. Math. Soc. Japan, 20, 1968
Keywords: Essential topologyNature: Original Retrieve article from Numdam
V: 30, 311-341, LNM 191 (1971)
WATANABE, Takesi
On balayées of excessive measures and functions with respect to resolvents (
Potential theory)
A general study of balayage of excessive measures as dual to réduite of excessive functions, first for a single kernel, then for a resolvent on a measurable space, and finally for a standard process
Comment: See Kunita and T. Watanabe,
Ill. J. Math.,
9, 1965. For the modern theory of balayage of measures (using Kuznetsov's processes) see Getoor,
Excessive Measures, 1990, Chapter 4
Keywords: Excessive measures,
BalayageNature: Original Retrieve article from Numdam
V: 31, 342-346, LNM 191 (1971)
WEIL, Michel
Décomposition d'un temps terminal (
Markov processes)
It is shown that for a Hunt process, a terminal time can be represented as the infimum of a previsible terminal time, and a totally inaccessible terminal time
Keywords: Terminal timesNature: Original Retrieve article from Numdam
V: 32, 347-361, LNM 191 (1971)
WEIL, Michel
Quasi-processus et énergie (
Markov processes,
Potential theory)
The energy of an excessive function $f$ with respect to an excessive measure $\xi$ has a simple proba\-bi\-listic interpretation if $\xi$ is is the potential of a measure $\mu$ and $f$ is the potential of an additive functional $(A_t)$, as ${1\over2}E_\mu[A_\infty^2]$. If $\xi$ is not a potential, still it can be associated with it a quasi-process (see Weil
418) with a birthtime $b$ and a death time $d$, and the formal expression ${1\over2}E[(A_d-A_b)^2]$ is given a precise meaning and represents the energy
Comment: This subject has been renewed by the introduction of Kuznetsov's measures. See Fitzsimmons
Sem. Stoch. Proc., 1987
Keywords: Hunt quasi-processes,
EnergyNature: Original Retrieve article from Numdam
V: 33, 362-372, LNM 191 (1971)
WEIL, Michel
Conditionnement par rapport au passé strict (
Markov processes)
Given a totally inaccessible terminal time $T$, it is shown how to compute conditional expectations of the future with respect to the strict past $\sigma$-field ${\cal F}_{T-}$. The formula involves the Lévy system of the process
Comment: B. Maisonneuve pointed out once that the paper, though essentially correct, has a small mistake somewhere. See Dellacherie-Meyer,
Probabilité et Potentiels, Chap. XX
46--48
Keywords: Terminal times,
Lévy systemsNature: Original Retrieve article from Numdam
VI: 01, 1-34, LNM 258 (1972)
ARTZNER, Philippe
Echantillons et couples indépendants de points aléatoires portés par une surface convexe (
Independence)
The general problem is to find conditions under which a convolution equation $\mu{*}\mu=\mu'{*}\mu''$ in $
R^n$ implies that $\mu'=\mu''=\mu$. It is shown here that the result is true if the measures are carried by a convex surface which is not too flat
Comment: The results were announced in the note
C. R. Acad. Sci., 272, 1971
Keywords: Decomposition of lawsNature: Original Retrieve article from Numdam
VI: 02, 35-50, LNM 258 (1972)
AZÉMA, Jacques
Une remarque sur les temps de retour. Trois applications (
Markov processes,
General theory of processes)
This paper is the first step in the investigations of Azéma on the ``dual'' form of the general theory of processes (for which see Azéma (
Ann. Sci. ENS, 6, 1973, and
814). Here the $\sigma$-fields of cooptional and coprevisible sets are introduced in a Markovian set-up, and without their definitive names. A section theorem by return times is proved, and applications to the theory of Markov processes are given
Keywords: Homogeneous processes,
Coprevisible processes,
Cooptional processes,
Section theorems,
Projection theorems,
Time reversalNature: Original Retrieve article from Numdam
VI: 03, 51-71, LNM 258 (1972)
BRETAGNOLLE, Jean
$p$-variation de fonctions aléatoires~: 1. Séries de Rademacher 2. Processus à accoissements indépendants (
Independent increments)
The main result of the paper is theorem III, which gives a necessary and sufficient condition for the sample paths of a centered Lévy process to have a.s. a finite $p$-variation on finite time intervals, for $1<p<2$: the process should have no Gaussian part, and $|x|^p$ be integrable near $0$ w.r.t. the Lévy measure $L(dx)$. The proof rests on discrete estimates on the $p$-variation of Rademacher series. Additional results on $h$-variation w.r.t. more general convex functions are given or mentioned
Comment: This paper improves on Millar,
Zeit. für W-theorie, 17, 1971
Keywords: $p$-variation,
Rademacher functionsNature: Original Retrieve article from Numdam
VI: 05, 90-97, LNM 258 (1972)
CHUNG, Kai Lai
On universal field equations (
General theory of processes)
There is a pun in the title, since ``field'' here is a $\sigma$-field and not a quantum field. The author proves useful results on the $\sigma$-fields ${\cal F}_{T-}$ and ${\cal F}_{T+}$ associated with an arbitrary random variable $T$ in the paper of Chung-Doob,
Amer. J. Math.,
87, 1965. As a corollary, he can prove easily that for a Hunt process, accessible = previsible
Keywords: FiltrationsNature: Original Retrieve article from Numdam
VI: 06, 98-100, LNM 258 (1972)
KAZAMAKI, Norihiko
Examples on local martingales (
Martingale theory)
Two simple examples are given, the first one concerning the filtration generated by an exponential stopping time, the second one showing that local martingales are not preserved under time changes (Kazamaki,
Zeit. für W-theorie, 22, 1972)
Keywords: Changes of time,
Local martingales,
Weak martingalesNature: Original Retrieve article from Numdam
VI: 07, 101-104, LNM 258 (1972)
KAZAMAKI, Norihiko
Krickeberg's decomposition for local martingales (
Martingale theory)
It is shown that a local martingale bounded in $L^1$ is a difference of two (minimal) positive local martingales
Keywords: Local martingales,
Krickeberg decompositionNature: Original Retrieve article from Numdam
VI: 08, 105-108, LNM 258 (1972)
KAZAMAKI, Norihiko
Note on a stochastic integral equation (
Stochastic calculus)
Though this paper has been completely superseded by the theory of stochastic differential equations with respect to semimartingales (see
1124), it has a great historical importance as the first step in this direction: the semimartingale involved is the sum of a locally square integrable martingale and a continuous increasing process
Comment: The author developed the subject further in
Tôhoku Math. J. 26, 1974
Keywords: Stochastic differential equationsNature: Original Retrieve article from Numdam
VI: 09, 109-112, LNM 258 (1972)
SAM LAZARO, José de;
MEYER, Paul-André
Un gros processus de Markov. Application à certains flots (
Markov processes)
In a vague but useful sense, a ``big'' process over a given process consists of random variables whose values are a part of the path of the original process (the best known example is the excursion process). Here it is shown how the past of a Markov process can be turned into a big (homogeneous) Markov process, and how its semigroup is computed using an idea of Dawson (
Trans. Amer. Math. Soc.,
131, 1968)
Comment: For a complete account of Dawson's formula, see Dellacherie-Meyer,
Probabilités et Potentiel, \no XIV.45
Keywords: Prediction theory,
Filtered flowsNature: Original Retrieve article from Numdam
VI: 10, 113-117, LNM 258 (1972)
MAISONNEUVE, Bernard
Topologies du type de Skorohod (
General theory of processes)
This paper presents an adaptation of the well known Skorohod topology, to the case of an arbitrary (i.e., non-compact) interval of the line
Keywords: Skorohod topologyNature: Original Retrieve article from Numdam
VI: 11, 118-129, LNM 258 (1972)
MEYER, Paul-André
La mesure de H. Föllmer en théorie des surmartingales (
Martingale theory)
The Föllmer measure of a supermartingale is an extension to very general situation of the construction of $h$-path processes in the Markovian case. Let $\Omega$ be a probability space with a filtration, let $\Omega'$ be the product space $[0,\infty]\times\Omega$, the added coordinate playing the role of a lifetime $\zeta$. Then the Föllmer measure associated with a supermartingale $(X_t)$ is a measure $\mu$ on this enlarged space which satisfies the property $\mu(]T,\infty])=E(X_T)$ for any stopping time $T$, and simple additional properties to ensure uniqueness. When $X_t$ is a class (D) potential, it turns out to be the usual Doléans measure, but except in this case its existence requires some measure theoretic conditions on $\Omega$; which are slightly different here from those used by Föllmer,
Zeit für X-theorie, 21, 1970
Keywords: Supermartingales,
Föllmer measuresNature: Exposition,
Original additions Retrieve article from Numdam
VI: 12, 130-150, LNM 258 (1972)
MEYER, Paul-André
Le schéma de remplissage en temps continu, d'après H. Rost (
Ergodic theory,
Potential theory)
The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as
523. Here following Rost himself (
Invent. Math., 14, 1971) the construction is extended to continuous time Markov processes. In the transient case, the results are translated in potential-theoretic language, and proved using techniques due to Mokobodzki. Then the general case follows from this result applied to a space-time extension of the semi-group
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Filling scheme,
Balayage of measures,
Skorohod imbeddingNature: Exposition,
Original additions Retrieve article from Numdam
VI: 14, 159-163, LNM 258 (1972)
MEYER, Paul-André
Temps d'arrêt algébriquement prévisibles (
General theory of processes)
The main results concern the natural filtration of a right continuous process taking values in a Polish spaces, and defined on a Blackwell space $\Omega$. Conditions are given on a process or a random variable on $\Omega$ which insure that it will be previsible or optional under any probability law on $\Omega$
Comment: The subject has been kept alive by Azéma, who used similar techniques in several papers
Keywords: Stopping times,
Previsible processesNature: Original Retrieve article from Numdam
VI: 15, 164-167, LNM 258 (1972)
MEYER, Paul-André
Une note sur le théorème du balayage de Hunt (
Markov processes,
Potential theory)
The theorem involved is the characterization of the réduite $P_A u$ of an excessive function $u$ on a set $A$ as equal (except on a well-defined semipolar set) to the infimum of the excessive functions that dominate $u$ on $A$. This theorem is slightly improved under the absolute continuity hypothesis. The proof rests on the following property of the fine topology: every (nearly Borel) finely closed set is contained in a fine $G_\delta$ which differs from it by a polar set
Keywords: Réduite,
Fine topology,
Absolute continuity hypothesisNature: Original Retrieve article from Numdam
VI: 16, 168-172, LNM 258 (1972)
MEYER, Paul-André;
WALSH, John B.
Un résultat sur les résolvantes de Ray (
Markov processes)
This is a complement to the authors' paper on Ray processes in
Invent. Math., 14, 1971: a lemma is proved on the existence of many martingales which are continuous whenever the process is continuous (a wrong reference for it was given in the paper). Then it is shown that the mapping $x\rightarrow P_x$ is continuous in the weak topology of measures, when the path space is given the topology of convergence in measure. Note that a correction is mentioned on the errata page of vol. VII
Comment: The idea of using the topology of convergence in measure on a path space turned out to be a fruitful idea; see Meyer and Zheng
Ann. Inst. Henri Poincaré 20, 1984
Keywords: Ray compactification,
Weak convergence of measuresNature: Original Retrieve article from Numdam
VI: 17, 173-176, LNM 258 (1972)
MOKOBODZKI, Gabriel
Pseudo-quotient de deux mesures par rapport à un cône de potentiels. Application à la dualité (
Potential theory)
The last four pages of this paper have been omitted by mistake, and appear in the following volume as
729. The general results concerning the axiomatically defined cones of potentials (see for instance the author's exposition in
Séminaire Bourbaki, 1969-70,
377) are quickly reviewed first, and then applied to the following problem concerning the potential kernel $V$ of a resolvent: given a pair of measures $\lambda\le\mu$ in the sense of balayage, then we have $\lambda V\le \mu V$ in the ordinary sense. The corresponding density (dominated by $1$) does not depend on the resolvent, but only on the potential cones of excessive functions and potentials associated with it, and a way to compute it is indicated
Keywords: Cones of potentialsNature: Original Retrieve article from Numdam
VI: 18, 177-197, LNM 258 (1972)
NAGASAWA, Masao
Branching property of Markov processes (
Markov processes)
To be completed
Keywords: Branching processesNature: Original Retrieve article from Numdam
VI: 19, 198-201, LNM 258 (1972)
RAO, Murali
Doob's decomposition and Burkholder's inequalities (
Martingale theory)
The ``Burkholder inequalities'' referred here are the weak-$L^1$ estimates for the supremum of a martingale transform and for the square function proved by Burkholder (
Ann. Math. Stat., 37, 1966) for $L^1$-bounded discrete time martingales. The original proof was quite sophisticated, while here these inequalities are deduced from an estimate on the (elementary) Doob decomposition of a discrete supermartingale
Comment: This little-known paper would probably deserve a modern translation in continuous time
Keywords: Burkholder inequalities,
Decomposition of supermartingalesNature: Original Retrieve article from Numdam
VI: 20, 202-214, LNM 258 (1972)
REVUZ, Daniel
Le principe semi-complet du maximum (
Potential theory)
The problem studied here (and not completely solved) consists in finding potential theoretic characterizations for the recurrent potential operators constructed in the basic paper of Neveu,
Ann. Inst. Fourier, 22-2, 1972. It is shown that these operators satisfy suitable maximum principles (as usual, slightly stronger in the discrete case than in the continuous case). The converse is delicate and some earlier work of Kondo (
Osaka J. Math.,
4, 1967) and Oshima (same journal,
6, 1969) is discussed in this new set-up
Comment: This topic is discussed again in Revuz' book
Markov Chains, North-Holland
Keywords: Recurrent potential theory,
Maximum principles,
Recurrent Markov chainsNature: Original Retrieve article from Numdam
VI: 21, 215-232, LNM 258 (1972)
WALSH, John B.
Transition functions of Markov processes (
Markov processes)
Assume that a cadlag process satisfies the strong Markov property with respect to some family of kernels $P_t$ (not necessarily a semigroup). It is shown that these kernels can be modified into a true strong Markov transition function with a few additional properties. A similar problem is solved for a left continuous, moderate Markov process. The technique involves a Ray compactification which is eliminated at the end, and a useful lemma shows how to construct supermedian functions which separate points
Comment: The problem discussed here has great theoretical importance, but little practical importance except for time reversal. The construction of a nice transition function for a Markov process has been also discussed by Kuznetsov ()
Keywords: Transition functions,
Strong Markov property,
Moderate Markov property,
Ray compactificationNature: Original Retrieve article from Numdam
VI: 22, 233-242, LNM 258 (1972)
WALSH, John B.
The perfection of multiplicative functionals (
Markov processes)
In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see
203) that every functional has a perfect modification, see also Dellacherie
304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure
Keywords: Multiplicative functionals,
Perfection,
Essential topologyNature: Original Retrieve article from Numdam
VI: 23, 243-252, LNM 258 (1972)
MEYER, Paul-André
Quelques autres applications de la méthode de Walsh (``La perfection en probabilités'') (
Markov processes)
This is but an exercise on using the method of the preceding paper
622 to reduce the exceptional sets in other situations: additive functionals, cooptional times and processes, etc
Comment: A correction to this paper is mentioned on the errata list of vol. VII
Keywords: Additive functionals,
Return times,
Essential topologyNature: Original Retrieve article from Numdam
VII: 01, 1-24, LNM 321 (1973)
BENVENISTE, Albert
Application de deux théorèmes de G.~Mokobodzki à l'étude du noyau de Lévy d'un processus de Hunt sans hypothèse (L) (
Markov processes)
The object of the theory of Lévy systems is to compute the previsible compensator of sums $\sum_{s\le t} f(X_{s-},X_s)$ extended to the jump times of a Markov process~$X$, i.e., the times $s$ at which $X_s\not=X_{s-}$. The theory was created by Lévy in the case of a process with independent increments, and the classical results for Markov processes are due to Ikeda-Watanabe,
J. Math. Kyoto Univ.,
2, 1962 and Watanabe,
Japan J. Math.,
34, 1964. An exposition of their results can be found in the Seminar,
106. The standard assumptions were: 1) $X$ is a Hunt process, implying that jumps occur at totally inaccessible stopping times and the compensator is continuous, 2) Hypothesis (L) (absolute continuity of the resolvent) is satisfied. Here using two results of Mokobodzki: 1) every excessive function dominated in the strong sense in a potential. 2) The existence of medial limits (this volume,
719), Hypothesis (L) is shown to be unnecessary
Comment: Mokobodzki's second result depends on additional axioms in set theory, the continuum hypothesis or Martin's axiom. See also Benveniste-Jacod,
Invent. Math. 21, 1973, which no longer uses medial limits
Keywords: Lévy systems,
Additive functionalsNature: Original Retrieve article from Numdam
VII: 02, 25-32, LNM 321 (1973)
MEYER, Paul-André
Une mise au point sur les systèmes de Lévy. Remarques sur l'exposé de A. Benveniste (
Markov processes)
This is an addition to the preceding paper
701, extending the theory to right processes by means of a Ray compactification
Comment: All this material has become classical. See for instance Dellacherie-Meyer,
Probabilités et Potentiel, vol. D, chapter XV, 31--35
Keywords: Lévy systems,
Ray compactificationNature: Original Retrieve article from Numdam
VII: 03, 33-35, LNM 321 (1973)
DELLACHERIE, Claude
Un crible généralisé (
Descriptive set theory)
Given a Borel set $A$ in the product $E\times F$ of two compact metric sets, the set of all $x\in E$ such that the section $A(x)\subset Y$ is of second category is analytic
Comment: The authour discovered later that the main result is in fact due to Novikov: two references are given in
1252Keywords: Analytic setsNature: Original Retrieve article from Numdam
VII: 04, 36-37, LNM 321 (1973)
DELLACHERIE, Claude
Temps d'arrêt totalement inaccessibles (
General theory of processes)
Given an accessible random set $H$ and a totally inaccessible stopping time $T$, whenever $T(\omega)\in H(\omega)$ then $T(\omega)$ is a condensation point of $H(\omega)$ on the left, i.e., there are uncountably many points of $H$ arbitrarily close to $T(\omega)$ on the left
Keywords: Stopping times,
Accessible sets,
Totally inaccessible stopping timesNature: Original Retrieve article from Numdam
VII: 05, 38-47, LNM 321 (1973)
DELLACHERIE, Claude
Sur les théorèmes fondamentaux de la théorie générale des processus (
General theory of processes)
This paper reconstructs the general theory of processes starting from a suitable family ${\cal V }$ of stopping times, and the $\sigma$-field generated by stochastic intervals $[S,T[$ with $S,T\in{\cal V }$, $S\le T$. Section and projection theorems are proved
Comment: The idea of this paper has proved fruitful. See for instance Lenglart,
1449; Le Jan,
Z. für W-Theorie 44, 1978
Keywords: Stopping times,
Section theoremsNature: Original Retrieve article from Numdam
VII: 07, 51-57, LNM 321 (1973)
DELLACHERIE, Claude
Une conjecture sur les ensembles semi-polaires (
Markov processes)
For a right process satisfying the absolute continuity hypothesis and assuming singletons are semi-polar sets, it is conjectured that a (nearly-)Borel set is semipolar if and only if it does not contain uncountable families of disjoint, non-polar compact sets. This statement implies that two processes which have the same polar sets also have the same semi-polar sets
Comment: The conjecture can be proved, using a general result of Mokobodzki, see
1238Keywords: Polar sets,
Semi-polar setsNature: Original Retrieve article from Numdam
VII: 08, 58-60, LNM 321 (1973)
DELLACHERIE, Claude
Potentiels de fonctionnelles additives. Un contre-exemple de Knight (
Markov processes)
An example is given of a Markov process and a continuous additive functional $(A_t)$ such that $A_{\infty}$ is finite, and whose potential is finite except at one single (polar) point
Keywords: Additive functionalsNature: Exposition,
Original additions Retrieve article from Numdam
VII: 09, 61-76, LNM 321 (1973)
FARAUT, Jacques
Fonction brownienne sur une variété riemannienne (
Miscellanea,
Gaussian processes)
As defined originally by Lévy in the case of spheres and euclidian spaces, a Brownian motion indexed by a point of a metric space $E$ is a centered Gaussian process $(X_t)_{t\in E}$ such that $E[(X_t-X_s)^2]=d(s,t)$, the distance. In a Riemannian manifold $d$ is understood to be the geodesic distance. The results of this paper imply that Brownian motions exist on spheres and Euclidean spaces (Lévy's original result), on real hyperbolic spaces, but not on quaternionic hyperbolic spaces
Comment: This article contains joint work with K. Harzallah
Keywords: Covariance,
Riemannian manifold,
Riemannian distance,
Lévy Brownian motions,
Several parameter Brownian motionsNature: Original Retrieve article from Numdam
VII: 10, 77-94, LNM 321 (1973)
HEINKEL, Bernard
Une condition suffisante pour la continuité presque sûre des trajectoires de certains processus gaussiens (
Gaussian processes)
It is shown that a continuity criterion due to Preston (1972) can be deduced from a theorem of Dudley (1967)
Comment: To be completed
Keywords: Continuity of paths of Gaussian processesNature: Original Retrieve article from Numdam
VII: 11, 95-117, LNM 321 (1973)
EL KAROUI, Nicole;
REINHARD, Hervé
Processus de diffusion dans ${\bf R}^n$ (
Diffusion theory)
This paper concerns diffusions (without boundaries) whose generators have Borel bounded coefficients. It consists of two parts. The first one is devoted to the equivalence between the existence and uniqueness of the diffusion semigroup and the uniqueness in law of the solution of the corresponding Ito stochastic differential equation. This allows the authors to use in the elliptic case the deep results of Krylov on s.d.e.'s. The second part concerns mostly the Lipschitz case, and discusses several properties of the diffusion process in itself: the representation of additive functional martingales; the relations between the number of martingales necessary for the representation and the rank of the generator (locally); the existence of a dual diffusion; the support and absolute continuity properties of the semi-group
Comment: This paper is in part an improved version of a paper on degenerate diffusions by Bonami, El-Karoui, Reinhard and Roynette (
Ann. Inst. H. Poincaré, 7, 1971)
Keywords: Construction of diffusions,
Diffusions with measurable coefficients,
Degenerate diffusionsNature: Original Retrieve article from Numdam
VII: 12, 118-121, LNM 321 (1973)
KAZAMAKI, Norihiko
Une note sur les martingales faibles (
Martingale theory)
Métivier has distinguished in the general theory of processes localization from prelocalization: a process $X$ is a local martingale if there exist stopping times $T_n$ increasing to infinity and martingales $M_n$ such that $X=M_n$ on the closed interval $[0,T_n]$ (omitting for simplicity the convention about time $0$). Replacing the closed intervals by open intervals $[0,T_n[$ defines prelocal martingales or
weak martingales. It is shown that in the filtration generated by one single stopping time, processes which are prelocally martingales (square integrable martingales) are so globally. It follows that prelocal martingales may not be prelocally square integrable
Comment: The interest of weak martingales arises from their invariance by (possibly discontinuous) changes of time, see Kazamaki,
Zeit. für W-theorie, 22, 1972
Keywords: Weak martingalesNature: Original Retrieve article from Numdam
VII: 13, 122-135, LNM 321 (1973)
KHALILI-FRANÇON, Elisabeth
Processus de Galton-Watson (
Markov processes)
This paper is mostly a survey of previous results with comments and some alternative proofs
Comment: An erroneous statement is corrected in
939Keywords: Branching processes,
Galton-Watson processesNature: Exposition,
Original additions Retrieve article from Numdam
VII: 14, 136-145, LNM 321 (1973)
MEYER, Paul-André
Le dual de $H^1$ est $BMO$ (cas continu) (
Martingale theory)
The basic results of Fefferman and Fefferman-Stein on functions of bounded mean oscillation in $
R$ and $
R^n$ and the duality between $BMO$ and $H^1$ were almost immediately translated into discrete martingale theory by Herz and Garsia. The next step, due to Getoor-Sharpe ({\sl Invent. Math.}
16, 1972), delt with continuous martingales. The extension to right continuous martingales, a good exercise in martingale theory, is given here
Comment: See
907 for a correction. This material has been published in book form, see for instance Dellacherie-Meyer,
Probabilités et Potentiel, Vol. B, Chapter VII
Keywords: $BMO$,
Hardy spaces,
Fefferman inequalityNature: Original Retrieve article from Numdam
VII: 16, 155-171, LNM 321 (1973)
MEYER, Paul-André;
TRAKI, Mohammed
Réduites et jeux de hasard (
Potential theory)
This paper arose from an attempt (by the second author) to rewrite the results of Dubins-Savage
How to Gamble if you Must in the language of standard (countably additive) measure theory, using the methods of descriptive set theory (analytic sets, section theorems, etc). The attempt is successful, since all general theorems can be proved in this set-up. More recent results in the same line, due to Strauch and Sudderth, are extended too. An appendix includes useful comments by Mokobodzki on the case of a gambling house consisting of a single kernel (discrete potential theory)
Comment: This material is reworked in Dellacherie-Meyer,
Probabilités et Potentiel, Vol. C, Chapter X
Keywords: Balayage,
Gambling house,
Réduite,
Optimal strategyNature: Original Retrieve article from Numdam
VII: 17, 172-179, LNM 321 (1973)
MEYER, Paul-André
Application de l'exposé précédent aux processus de Markov (
Markov processes)
This paper is devoted to results of J.F. Mertens on optimal stopping and strongly supermedian functions for a right Markov process (
Zeit. für W-theorie, 26, 1973), which are shown to be closely related to those of the preceding paper
716 on general gambling houses. An interesting result of Mokobodzki is included, showing that the extreme points of the convex set of all balayées of a given measure $\lambda$ are the balayées of $\lambda$ on sets
Comment: See related papers by Mertens in
Zeit. für W-theorie, 22, 1972 and
Invent. Math.,
23, 1974. The original result of Mokobodzki appeared in the
Sémin. Théorie du Potentiel, 1969-70
Keywords: Excessive functions,
Supermedian functions,
RéduiteNature: Exposition,
Original proofs Retrieve article from Numdam
VII: 20, 205-209, LNM 321 (1973)
MEYER, Paul-André
Remarques sur les hypothèses droites (
Markov processes)
The following technical point is discussed: why does one assume among the ``right hypotheses'' that excessive functions are nearly-Borel?
Keywords: Right processes,
Excessive functionsNature: Original Retrieve article from Numdam
VII: 21, 210-216, LNM 321 (1973)
MEYER, Paul-André
Note sur l'interprétation des mesures d'équilibre (
Markov processes)
Let $(X_t)$ be a transient Markov process (we omit the detailed assumptions) with a potential density $u(x,y)$. Let $\mu$ be the measure whose potential is the equilibrium potential of a set $A$. Then the distribution of the process at the last exit time from $A$ is given by $$E_x[f\circ X_{L-}, 0<L<\infty]=\int u(x,y)\,f(y)\,\mu(dy)$$ This formula, due to Chung, is deduced under minimal duality hypotheses from a general formula of Azéma, and a well-known theorem on Revuz measures
Keywords: Equilibrium potentials,
Last exit time,
Revuz measuresNature: Exposition,
Original proofs Retrieve article from Numdam
VII: 22, 217-222, LNM 321 (1973)
MEYER, Paul-André
Sur les désintégrations régulières de L. Schwartz (
General theory of processes)
This paper presents a small part of an important article of L.~Schwartz (
J. Anal. Math.,
26, 1973). The main result is an extension of the classical theorem on the existence of regular conditional probability distributions: on a good filtered probability space, the previsible and optional projections of a process can be computed by means of true kernels
Comment: The contents of this paper have been considerably developed by F.~Knight in his theory of prediction. See
1007Keywords: Previsible projections,
Optional projections,
Prediction theoryNature: Exposition,
Original additions Retrieve article from Numdam
VII: 24, 248-272, LNM 321 (1973)
MÜRMANN, Michael G.
A semi-Markovian model for the Brownian motion (
Statistical mechanics)
A model for physical Brownian motion (the effect on a heavy particle of many interactions with light particles), originally proposed by Spitzer and Holley, in dimension 1, is studied in detail. The resulting process, whose construction is delicate, is non-Markovian
Comment: The last two pages of the manuscript (proof of Proposition 5 and References) were omitted at the production stage, and added as a loose sheet to vol. VIII, while another loose sheet contains an example. These sheets are not mentioned in the table of contents of vol. VIII
Keywords: Infinite particle systemsNature: Original Retrieve article from Numdam
VII: 26, 284-290, LNM 321 (1973)
ROST, Hermann
Relaxation in infinite spin systems (
Statistical mechanics)
The existence of a stochastic process describing an infinitely many interacting particle system is proved
Comment: This is related to Sullivan,
Zeit. für W-theorie, 31, 1974
Keywords: Interacting particle systemsNature: Original Retrieve article from Numdam
VII: 27, 291-300, LNM 321 (1973)
TAYLOR, John C.
On the existence of resolvents (
Potential theory)
Since the basic results of Hunt, a kernel satisfying the complete maximum principle is expected to be the potential kernel of a sub-Markov resolvent. This is not always the case, however, and one should also express that, so to speak, ``potentials vanish at the boundary''. Such a condition is given here on an abstract space, which supersedes an earlier result of the author (
Invent. Math. 17, 1972) and a result of Hirsch (
Ann. Inst. Fourier, 22-1, 1972)
Comment: The definitive paper of Taylor on this subject appeared in
Ann. Prob.,
3, 1975
Keywords: Complete maximum principle,
ResolventsNature: Original Retrieve article from Numdam
VII: 28, 301-318, LNM 321 (1973)
WALDENFELS, Wilhelm von
Some remarks on Burkhardt's model for pressure broadening of spectral lines (
Miscellanea)
A model proposed by Burkhardt in 1940 for the deformation of the radiation emitted by an atom due to the surrounding atoms is transformed into probabilistic language and exactly solved
Nature: Original Retrieve article from Numdam
VIII: 01, 1-10, LNM 381 (1974)
AZÉMA, Jacques;
MEYER, Paul-André
Une nouvelle représentation du type de Skorohod (
Markov processes)
A Skorohod imbedding theorem for general Markov processes is proved, in which the stopping time is a randomized ``left'' terminal time. A uniqueness result is proved
Comment: The result is deduced from a representation of measures by left additive functionals, due to Azéma (
Invent. Math. 18, 1973 and this volume,
814). A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbedding,
Multiplicative functionalsNature: Original Retrieve article from Numdam
VIII: 02, 11-19, LNM 381 (1974)
BRETAGNOLLE, Jean
Une remarque sur le problème de Skorohod (
Brownian motion)
The explicit construction of a non-randomized solution of the Skorohod imbedding problem given by Dubins (see
516) is studied from the point of view of exponential moments. In particular, the Dubins stopping time for the distribution of a bounded stopping time $T$ has exponential moments, but this is not always the case if $T$ has exponential moments without being bounded
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
VIII: 03, 20-21, LNM 381 (1974)
CHUNG, Kai Lai
Note on last exit decomposition (
Markov processes)
This is a useful complement to the monograph of Chung
Lectures on Boundary Theory for Markov Chains, Annals of Math. Studies 65, Princeton 1970
Keywords: Markov chainsNature: Original Retrieve article from Numdam
VIII: 04, 22-24, LNM 381 (1974)
DELLACHERIE, Claude
Un ensemble progressivement mesurable... (
General theory of processes)
The set of starting times of Brownian excursions from $0$ is a well-known example of a progressive set which does not contain any graph of stopping time. Here it is shown that considering the same set for the excursions from any $a$ and taking the union of all $a$, the corresponding set has the same property and has uncountable sections
Comment: Other such examples are known, such as the set of times at which the law of the iterated logarithm fails
Keywords: Progressive sets,
Section theoremsNature: Original Retrieve article from Numdam
VIII: 05, 25-26, LNM 381 (1974)
DELLACHERIE, Claude
Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (
General theory of processes)
This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems
Comment: A gap in the proof is filled in
928 and
2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor,
Z. für W-theorie, 38, 1977 and Yor
1221. For another approach to the restricted case considered here, see Ruiz de Chavez
1821. The previsible representation property of Brownian motion and compensated Poisson process was know by Itô; it is a consequence of the (stronger) chaotic representation property, established by Wiener in 1938. The converse was also known by Itô: among the martingales which are also Lévy processes, only Brownian motions and compensated Poisson processes have the previsible representation property
Keywords: Brownian motion,
Poisson processes,
Previsible representationNature: Original Retrieve article from Numdam
VIII: 06, 27-36, LNM 381 (1974)
DINGES, Hermann
Stopping sequences (
Markov processes,
Potential theory)
Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's ``filling scheme'', and several operations on stopping sequences, aiming at the construction of ``short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process
Comment: This is a development of the research of H.~Rost on the ``filling scheme'', for which see
523,
524,
612. This article contains announcements of further results
Keywords: Discrete time Markov processes,
Skorohod imbedding,
Filling schemeNature: Original Retrieve article from Numdam
VIII: 07, 37-77, LNM 381 (1974)
DUPUIS, Claire
Mesure de Hausdorff de la trajectoire de certains processus à accroissements indépendants et stationnaires (
Independent increments)
The problem is to show that, for symmetric Lévy processes with small jumps and without a Brownian part, there exists a natural Hausdorff measure for which almost every path up to time $t$ has ``length'' exactly $t$. The case of Brownian motion had been known for a long time, the case of stable processes was settled by S.J.~Taylor (
J. Math. Mech.,
16, 1967) whose methods are generalized here
Keywords: Hausdorff measures,
Lévy processesNature: Original Retrieve article from Numdam
VIII: 08, 78-79, LNM 381 (1974)
FERNIQUE, Xavier
Une démonstration simple du théorème de R.M.~Dudley et M.~Kanter sur les lois 0-1 pour les mesures stables (
Miscellanea)
The theorem concerns stable laws on a linear space, and asserts that every measurable linear subspace has probability 0 or 1. The title describes accurately the paper
Keywords: Stable measuresNature: Original Retrieve article from Numdam
VIII: 09, 80-133, LNM 381 (1974)
GEBUHRER, Marc Olivier
Une classe de processus de Markov en mécanique relativiste. Laplaciens généralisés sur les espaces symétriques de type non compact (
Markov processes)
The first part of this paper is devoted to a model of relativistic Brownian motion defined by Dudley (
Arkiv för Math.,
6, 1965-67), which is studied as a Lorentz invariant diffusion process (in the usual sense) on the standard hyperboloid of velocities in special relativity, on which the Lorentz group acts. The Brownian paths themselves are constructed by integration and possess a speed smaller than the velocity of light but no higher derivatives. The second part studies more generally invariant Markov processes on a Riemannian symmetric space of non-compact type, their generators and the corresponding semigroups
Keywords: Relativistic Brownian motion,
Invariant Markov processes,
Symmetric spacesNature: Exposition,
Original additions Retrieve article from Numdam
VIII: 10, 134-149, LNM 381 (1974)
KNIGHT, Frank B.
Existence of small oscillations at zeros of brownian motion (
Brownian motion)
The one-dimensional Brownian motion path is shown to have an abnormal behaviour (an ``iterated logarithm'' upper limit smaller than one) at uncountably many times on his set of zeros
Comment: This result may be compared to Kahane,
C.R. Acad. Sci. 248, 1974
Keywords: Law of the iterated logarithm,
Local timesNature: Original Retrieve article from Numdam
VIII: 11, 150-154, LNM 381 (1974)
HEATH, David C.
Skorohod stopping via Potential Theory (
Potential theory,
Markov processes)
The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $
R^n$ is given. A theorem of Mokobodzki on réduites is used as an important technical tool
Comment: This paper is best read in connection with
931. A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
VIII: 12, 155-171, LNM 381 (1974)
HEINKEL, Bernard
Théorèmes de dérivation du type de Lebesgue et continuité presque sûre de certains processus gaussiens (
Gaussian processes)
To be completed
Comment: To be completed
Keywords: Continuity of paths of Gaussian processesNature: Original Retrieve article from Numdam
VIII: 13, 172-261, LNM 381 (1974)
MAISONNEUVE, Bernard;
MEYER, Paul-André
Ensembles aléatoires markoviens homogènes (5 talks) (
Markov processes)
This long exposition is a development of original work by the first author. Its purpose is the study of processes which possess a strong Markov property, not at all stopping times, but only at those which belong to a given homogeneous random set $M$---a point of view introduced earlier in renewal theory (Kingman, Krylov-Yushkevich, Hoffmann-Jörgensen, see
412). The first part is devoted to technical results: the description of (closed) optional random sets in the general theory of processes, and of the operations of balayage of random measures; homogeneous processes, random sets and additive functionals; right Markov processes and the perfection of additive functionals. This last section is very technical (a general problem with this paper).\par Chapter II starts with the classification of the starting points of excursions (``left endpoints'' below) from a random set, and the fact that the projection (optional and previsible) of a raw AF still is an AF. The main theorem then computes the $p$-balayage on $M$ of an additive functional of the form $A_t=\int_0^th\circ X_s ds$. All these balayages have densities with respect to a suitable local time of $M$, which can be regularized to yield a resolvent and then a semigroup. Then the result is translated into the language of homogeneous random measures carried by the set of left endpoints and describing the following excursion. This section is an enlarged exposition of results due to Getoor-Sharpe (
Ann. Prob. 1, 1973;
Indiana Math. J. 23, 1973). The basic and earlier paper of Dynkin on the same subject (
Teor. Ver. Prim. 16, 1971) was not known to the authors.\par Chapter III is devoted to the original work of Maisonneuve on incursions. Roughly, the incursion at time $t$ is trivial if $t\in M$, and if $t\notin M$ it consists of the post-$t$ part of the excursion straddling $t$. Thus the incursion process is a path valued, non adapted process. It is only adapted to the filtration ${\cal F}_{D_t}$ where $D_t$ is the first hitting time of $M$ after $t$. Contrary to the Ito theory of excursions, no change of time using a local time is performed. The main result is the fact that, if a suitable regeneration property is assumed only on the set $M$ then, in a suitable topology on the space of paths, this process is a right-continuous strong Markov process. Considerable effort is devoted to proving that it is even a right process (the technique is heavy and many errors have crept in, some of them corrected in
932-
933).\par Chapter IV makes the connection between II and III: the main results of Chapter II are proved anew (without balayage or Laplace transforms): they amount to computing the Lévy system of the incursion process. Finally, Chapter V consists of applications, among which a short discussion of the boundary theory for Markov chains
Comment: This paper is a piece of a large literature. Some earlier papers have been mentioned above. Maisonneuve published as
Systèmes Régénératifs, Astérisque, 15, 1974, a much simpler version of his own results, and discovered important improvements later on (some of which are included in Dellacherie-Maisonneuve-Meyer,
Probabilités et Potentiel, Chapter XX, 1992). Along the slightly different line of Dynkin, see El~Karoui-Reinhard,
Compactification et balayage de processus droits, Astérisque 21, 1975. A recent book on excursion theory is Blumenthal,
Excursions of Markov Processes, Birkhäuser 1992
Keywords: Regenerative systems,
Regenerative sets,
Renewal theory,
Local times,
Excursions,
Markov chains,
IncursionsNature: Original Retrieve article from Numdam
VIII: 14, 262-288, LNM 381 (1974)
MEYER, Paul-André
Les travaux d'Azéma sur le retournement du temps (
General theory of processes,
Markov processes)
This paper is an exposition of a paper by Azéma (
Ann. Sci. ENS, 6, 1973) in which the theory ``dual'' to the general theory of processes was developed. It is shown first how the general theory itself can be developed from a family of killing operators, and then how the dual theory follows from a family of shift operators $\theta_t$. A transience hypothesis involving the existence of many ``return times'' permits the construction of a theory completely similar to the usual one. Then some of Azéma's applications to the theory of Markov processes are given, particularly the representation of a measure not charging $\mu$-polar sets as expectation under the initial measure $\mu$ of a left additive functional
Comment: This paper follows (with considerable progress) the line of
602. The names given by Azéma to right and left additive functionals are exchanged. Another difference with Azéma's original paper is the fact that the lifetime $\zeta$ does not appear. All these results have been included in Dellacherie-Maisonneuve-Meyer,
Probabilités et Potentiel, Chapter XVIII, 1992
Keywords: Time reversal,
Shift operators,
Killing operators,
Cooptional processes,
Coprevisible processes,
Additive functionals,
Left additive functionalsNature: Exposition,
Original additions Retrieve article from Numdam
VIII: 15, 289-289, LNM 381 (1974)
MEYER, Paul-André
Une note sur la compactification de Ray (
Markov processes)
This short note shows that (contrary to the belief of Meyer and Walsh in a preceding paper) the state space of a Ray process is universally measurable in its Ray compactification
Comment: This is now considered a standard fact
Keywords: Ray compactification,
Right processesNature: Original Retrieve article from Numdam
VIII: 17, 310-315, LNM 381 (1974)
MEYER, Paul-André
Une représentation de surmartingales (
Martingale theory)
Garsia asked whether every right continuous positive supermartingale $(X_t)$ bounded by $1$ is the optional projection of a (non-adapted) decreasing process $(D_t)$, also bounded by $1$. This problem is solved by an explicit formula, and a proof is sketched showing that, if boundedness is not assumed, the proper condition is $D_t\le X^{*}$
Comment: The ``exponential formula'' appearing in this paper was suggested by a more concrete problem in the theory of Markov processes, using a terminal time. Similar looking formulas occurs in multiplicative decompositions and in
801. For the much more difficult case of positive submartingales, see
1023 and above all Azéma,
Z. für W-theorie, 45, 1978 and its exposition
1321Keywords: Supermartingales,
Multiplicative decompositionNature: Original Retrieve article from Numdam
VIII: 18, 316-328, LNM 381 (1974)
PRIOURET, Pierre
Construction de processus de Markov sur ${\bf R}^n$ (
Markov processes,
Diffusion theory)
The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic
Comment: To be completed
Keywords: Processes with jumpsNature: Original Retrieve article from Numdam
VIII: 19, 329-343, LNM 381 (1974)
SMYTHE, Robert T.
Remarks on the hypotheses of duality (
Markov processes)
To develop the full strength of potential theory, it is helpful to assume that the basic Markov process has a right-continuous, strong Markov dual. The paper investigates what remains if this assumption is not made, i.e., if the process (transient and satisfying the absolute continuity hypothesis (hypothesis (L)) only has a left continuous moderately Markov dual process (Smythe-Walsh ,
Invent. Math.,
19, 1973)
Comment: The independent paper Garsia Alvarez-Meyer
Ann. Prob. 1, 1973, has some results in common with this one
Keywords: Dual semigroupsNature: Original Retrieve article from Numdam
VIII: 20, 344-354, LNM 381 (1974)
WALDENFELS, Wilhelm von
Taylor expansion of a Poisson measure (
Miscellanea)
To be completed
Comment: To be completed
Keywords: Poisson point processesNature: Original Retrieve article from Numdam
IX: 01, 2-96, LNM 465 (1975)
MEYER, Paul-André;
SAM LAZARO, José de
Questions de théorie des flots (7 chapters) (
Ergodic theory)
This is part of a seminar given in the year 1972/73. A flow is meant to be a one-parameter group $(\theta_t)$ of 1--1 measure preserving transformations of a probability space. The main topic of this seminar is the theory of filtered flows, i.e., a filtration $({\cal F}_t)$ ($t\!\in\!
R$) is given such that $\theta_s ^{-1}{\cal F}_t={\cal F}_{s+t}$, and particularly the study of
helixes, which are real valued processes $(Z_t)$ ($t\!\in\!
R$) such that $Z_0=0$, which for $t\ge0$ are adapted, and on the whole line have homogeneous increments ($Z_{s+t}-Z_t=Z_t\circ \theta_s$). Two main classes of helixes are considered, the increasing helixes, and the martingale helixes. Finally, a filtered flow such that ${\cal F}_{-\infty}$ is degenerate is called a K-flow (K for Kolmogorov). Chapter~1 gives these definitions and their simplest consequences, as well as the definition of (continuous time) point processes, and the Ambrose construction of (unfiltered) flows from discrete flows as
flows under a function. Chapter II shows that homogeneous discrete point processes and flows under a function are two names for the same object (Hanen,
Ann. Inst. H. Poincaré, 7, 1971), leading to the definition of the Palm measure of a discrete point process, and proves the classical (Ambrose-Kakutani) result that every flow with reasonable ergodicity properties can be interpreted as a flow under a function. A discussion of the case of filtered flows follows, with incomplete results. Chapter III is devoted to examples of flows and K-flows (Totoki's theorem). Chapter IV contains the study of increasing helixes, their Palm measures, and changes of times on flows. Chapter V is the original part of the seminar, devoted to the (square integrable) martingale helixes, their brackets, and the fact that in every K-flow these martingale helixes generate all martingales by stochastic integration. The main tool to prove this is a remark that every filtered K-flow can be interpreted (in a somewhat loose sense) as the flow of a stationary Markov process, helixes then becoming additive functionals, and standard Markovian methods becoming applicable. Chapter VI is devoted to spectral multiplicity, the main result being that a filtered flow, whenever it possesses one martingale helix, possesses infinitely many orthogonal helixes (orthogonal in a weak sense, not as martingales). Chapter VII is devoted to an independent topic: approximation in law of any ergodic stationary process by functionals of the Brownian flow (Nisio's theorem)
Comment: This set of lectures should be completed by the paper of Benveniste
902 which follows it, by an (earlier) paper by Sam Lazaro-Meyer (
Zeit. für W-theorie, 18, 1971) and a (later) paper by Sam Lazaro (
Zeit. für W-theorie, 30, 1974). Some of the results presented were less original than the authors believed at the time of the seminar, and due acknowledgments of priority are given; for an additional one see
1031. Related papers are due to Geman-Horowitz (
Ann. Inst. H. Poincaré, 9, 1973). The theory of filtered flows and Palm measures had a striking illustration within the theory of Markov processes as Kuznetsov measures (Kuznetsov,
Th. Prob. Appl.,
18, 1974) and the interpretation of ``Hunt quasi-processes'' as their Palm measures (Fitzsimmons,
Sem. Stoch. Processes 1987, 1988)
Keywords: Filtered flows,
Kolmogorov flow,
Flow under a function,
Ambrose-Kakutani theorem,
Helix,
Palm measuresNature: Exposition,
Original additions Retrieve article from Numdam
IX: 02, 97-153, LNM 465 (1975)
BENVENISTE, Albert
Processus stationnaires et mesures de Palm du flot spécial sous une fonction (
Ergodic theory,
General theory of processes)
This paper takes over several topics of
901, with important new results and often with simpler proofs. It contains results on the existence of ``perfect'' versions of helixes and stationary processes, a better (uncompleted) version of the filtration itself, a more complete and elegant exposition of the Ambrose-Kakutani theorem, taking the filtration into account (the fundamental counter is adapted). The general theory of processes (projection and section theorems) is developed for a filtered flow, taking into account the fact that the filtrations are uncompleted. It is shown that any bounded measure that does not charge ``polar sets'' is the Palm measure of some increasing helix (see also Geman-Horowitz (
Ann. Inst. H. Poincaré, 9, 1973). Then a deeper study of flows under a function is performed, leading to section theorems of optional or previsible homogeneous sets by optional or previsible counters. The last section (written in collaboration with J.~Jacod) concerns a stationary counter (discrete point process) in its natural filtration, and its stochastic intensity: here it is shown (contrary to the case of processes indexed by a half-line) that the stochastic intensity does not determine the law of the counter
Keywords: Filtered flows,
Flow under a function,
Ambrose-Kakutani theorem,
Helix,
Palm measures,
Perfection,
Point processesNature: Original Retrieve article from Numdam
IX: 03, 154-205, LNM 465 (1975)
NANOPOULOS, Photius
Mesures d'information et représentation de semi-groupes associés (
Miscellanea)
Several attempts have been made to define the information content of an event on a measurable space independently from its probability. This paper develops a point of view of Kampé de Fériet-Forte,
C.R.A.S. Paris, 265A, 1969
Keywords: Information theoryNature: Original Retrieve article from Numdam
IX: 05, 213-225, LNM 465 (1975)
CHOU, Ching Sung
Les méthodes d'A. Garsia en théorie des martingales. Extension au cas continu (
Martingale theory)
The methods developed in discrete time by Garsia
Martingale Inequalities: Seminar Notes on Recent Progress, Benjamin, 1973, are extended to continuous time
Comment: See Lenglart-Lépingle-Pratelli
1404. These methods have now become standard, and can be found in a number of books
Keywords: Inequalities,
Burkholder inequalitiesNature: Original Retrieve article from Numdam
IX: 06, 226-236, LNM 465 (1975)
CHOU, Ching Sung;
MEYER, Paul-André
Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels (
General theory of processes)
Dellacherie has studied in
405 the filtration generated by a point process with one single jump. His study is extended here to the filtration generated by a discrete point process. It is shown in particular how to construct a martingale which has the previsible representation property
Comment: In spite or because of its simplicity, this paper has become a standard reference in the field. For a general account of the subject, see He-Wang-Yan,
Semimartingale Theory and Stochastic Calculus, CRC~Press 1992
Keywords: Point processes,
Previsible representationNature: Original Retrieve article from Numdam
IX: 08, 239-245, LNM 465 (1975)
DELLACHERIE, Claude;
MEYER, Paul-André
Un nouveau théorème de projection et de section (
General theory of processes)
Optional section and projection theorems are proved without assuming the ``usual conditions'' on the filtration
Comment: This paper is obsolete. As stated at the end by the authors, the result could have been deduced from the general theorem in Dellacherie
705. The result takes its definitive form in Dellacherie-Meyer,
Probabilités et Potentiel, theorems IV.84 of vol. A and App.1, \no~6
Keywords: Section theorems,
Optional processes,
Projection theoremsNature: Original Retrieve article from Numdam
IX: 09, 246-267, LNM 465 (1975)
DACUNHA-CASTELLE, Didier
Processus et espaces de Banach invariants par réarrangement (
Miscellanea)
The first topic of this paper is the class of processes on an interval with exchangeable increments, which includes all processes with independent and stationary increments, but also the standard Brownian bridge (for several results quoted here, see Kallenberg,
Z. für W-theorie, 27, 1973). The results are applied to problems in the classification of Banach spaces, in the spirit of Dacunha-Castelle-Schreiber,
Ann. Inst. H. Poincaré, 10, 1974
Keywords: Banach spaces,
Exchangeable random variablesNature: Original Retrieve article from Numdam
IX: 10, 268-284, LNM 465 (1975)
DACUNHA-CASTELLE, Didier
Sous-espaces symétriques des espaces d'Orlicz (
Miscellanea)
Will be asked from the author
Comment: To be asked from the author. See Dacunha-Castelle-Schreiber,
Ann. Inst. H. Poincaré, 10, 1974
Keywords: Banach spacesNature: Original Retrieve article from Numdam
IX: 11, 285-293, LNM 465 (1975)
ARTZNER, Philippe
Quelques résultats de décomposabilité en algèbre linéaire et en algèbre quadratique aléatoires (
Miscellanea)
It is shown that if a random quadratic form on $
R^4$ is a.s. of rank 3 and has an absolutely continuous law, then its law is indecomposable. To prove this, it is shown that the sum of two independent planes in $
R^4$ cannot at the same time be a.s. a hyperplane and have an absolutely continuous law (with respect to the natural measure for hyperplanes)
Keywords: Independent random subspaces,
Independent quadratic formsNature: Original Retrieve article from Numdam
IX: 13, 305-317, LNM 465 (1975)
FÖLLMER, Hans
Phase transition and Martin boundary (
Miscellanea)
To be completed
Comment: To be completed
Keywords: Random fields,
Martin boundaryNature: Original Retrieve article from Numdam
IX: 14, 318-335, LNM 465 (1975)
FERNIQUE, Xavier
Des résultats nouveaux sur les processus gaussiens (
Gaussian processes)
Given a centered Gaussian process indexed by an arbitrary set~$T$, a major problem has been to find conditions implying that the sample functions are a.s. bounded, or a.s. continuous in the natural metric associated with the covariance. Here new necessary conditions for boundedness are given, which turn out to be sufficient in the case of stationary processes on $
R^n$. The conditions given here involve the existence of a majorizing measure, an idea which became crucial in the theory
Comment: For a systematic account of the theory around the time this paper was written, see Fernique's lectures in
École d'Été de Saint-Four~IV, LNM
480, 1974. For the definitive solution, see chapter 11 of Ledoux-Talagrand
Probability in Banach spaces, Springer 1991
Keywords: Gaussian processes,
Sample path regularityNature: Original Retrieve article from Numdam
IX: 16, 373-389, LNM 465 (1975)
DELLACHERIE, Claude;
MEYER, Paul-André
Ensembles analytiques et temps d'arrêt (
Descriptive set theory)
This is a sequel to the preceding paper
915. Instead of using the language of trees to prove the second separation theorem, a language more familiar to probabilists is used, in which the space of stopping times on $
N^
N$ is given a compact metric topology and the space of non-finite stopping times appears as the universal analytic, non-Borel set, from which all analytic sets can be constructed. Many proofs become very natural in this language
Comment: See also the next paper
917, the set of lectures by Dellacherie in C.A. Rogers,
Analytic Sets, Academic Press 1981, and chapter XXIV of Dellacherie-Meyer,
Probabilités et potentiel Keywords: Second separation theorem,
Stopping timesNature: Original Retrieve article from Numdam
IX: 17, 390-405, LNM 465 (1975)
DELLACHERIE, Claude
Jeux infinis avec information complète et temps d'arrêt (
Descriptive set theory)
This is a sequel to the preceding paper
916. It shows how well the language of stopping times applies, not only to the second separation theorem, but to the Gale-Stewart theorem on the determinacy of open games
Comment: The original remark on the relation between game determinacy and separation theorems, due to Blackwell (1967), led to a huge literature. More details can be found in chapter XXIV of Dellacherie-Meyer,
Probabilités et potentiel Keywords: Determinacy of games,
Gale and Stewart theoremNature: Original Retrieve article from Numdam
IX: 19, 408-419, LNM 465 (1975)
STRICKER, Christophe
Mesure de Föllmer en théorie des quasimartingales (
Martingale theory)
The Föllmer measure associated with a positive supermartingale, or more generally a quasimartingale (Föllmer,
Z. für W-theorie, 21, 1972;
Ann. Prob. 1, 1973) is constructed using a weak limit procedure instead of a projective limit
Comment: On Föllmer measures see
611. This paper corresponds to an early stage in the theory of quasimartingales, for which the main reference was Orey,
Proc. Fifth Berkeley Symp.,
2Keywords: Quasimartingales,
Föllmer measuresNature: Original Retrieve article from Numdam
IX: 20, 420-424, LNM 465 (1975)
STRICKER, Christophe
Une caractérisation des quasimartingales (
Martingale theory)
An integral criterion is shown to be equivalent to the usual definition of a quasimartingale using the stochastic variation
Keywords: QuasimartingalesNature: Original Retrieve article from Numdam
IX: 22, 437-442, LNM 465 (1975)
MOKOBODZKI, Gabriel
Relèvement borélien compatible avec une classe d'ensembles négligeables. Application à la désintégration des mesures (
Measure theory)
This is a beautiful application of the continuum ``hypothesis'' (axiom). It is shown that if $(E, {\cal E})$ is a separable measurable space (or more generally ${\cal E}$ has the power of the continuum) and ${\cal N}$ is a family of negligible sets within ${\cal E}$, then the space of classes of bounded measurable functions mod.~${\cal N}$ has a linear, isometric, and multiplicative lifting. The proof is rather simple. Essentially the same theorem was discovered independently by Chatterji, see
Vector and Operator Valued Measures, Academic Press 1973
Comment: The same proof leads to a slightly stronger and useful result (Meyer
2711): if $E$ is a compact metric space and if any two continuous functions equal a.e. are equal everywhere, the lifting can be taken to be the identity on continuous functions, and to be local, i.e., the liftings of two Borel functions equal a.e. in an open set are equal everywhere in this set
Keywords: Continuum axiom,
Lifting theorems,
Negligible setsNature: Original Retrieve article from Numdam
IX: 23, 443-463, LNM 465 (1975)
GETOOR, Ronald K.
On the construction of kernels (
Measure theory)
Given two measurable spaces $(E, {\cal E})$ $(F, {\cal F})$ and a family ${\cal N}\subset{\cal E}$ of negligible sets, a pseudo-kernel $T$ is a mapping from bounded measurable functions on $F$ to classes mod.${\cal N}$ of bounded measurable functions on $E$, which has all a.e. the properties (positivity, countable additivity) of a kernel. Regularizing $T$ consists in finding a true kernel $\hat T$ such that $\hat Tf$ belongs to the class $Tf$ for every measurable bounded $f$ on $F$. The regularization is easy whenever $F$ is compact metric. Then the result is extended to the case of a Lusin space, and to the case of a U-space (Radon space) assuming ${\cal N}$ consists of the negligible sets for a family of measures on $E$. An application is given to densities of continuous additive functionals of a Markov process
Comment: The author states that his paper is purely expository. This is not true, though the proof is a standard one in the theory of conditional distributions. For a deeper result, see Dellacherie
1030. For a presentation in book form, see Dellacherie-Meyer,
Probabilités et Potentiel C, chapter XI
41Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
IX: 24, 464-465, LNM 465 (1975)
MEYER, Paul-André
Une remarque sur la construction de noyaux (
Measure theory)
With the notation of the preceding report
923, this is a first attempt to solve the case (important in practice) where $F$ is coanalytic, assuming ${\cal N}$ consists of the negligible sets of a Choquet capacity
Comment: See Dellacherie
1030Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
IX: 25, 466-470, LNM 465 (1975)
MEYER, Paul-André;
YAN, Jia-An
Génération d'une famille de tribus par un processus croissant (
General theory of processes)
The previsible and optional $\sigma$-fields of a filtration $({\cal F}_t)$ on $\Omega$ are studied without the usual hypotheses: no measure is involved, and the filtration is not right continuous. It is proved that if the $\sigma$-fields ${\cal F}_{t-}$ are separable, then so is the previsible $\sigma$-field, and the filtration is the natural one for a continuous strictly increasing process. A similar result is proved for the optional $\sigma$-field assuming $\Omega$ is a Blackwell space, and then every measurable adapted process is optional
Comment: Making the filtration right continuous generally destroys the separability of the optional $\sigma$-field
Keywords: Previsible processes,
Optional processesNature: Original Retrieve article from Numdam
IX: 26, 471-485, LNM 465 (1975)
NAGASAWA, Masao
Multiplicative excessive measures and duality between equations of Boltzmann and of branching processes (
Markov processes,
Statistical mechanics)
The author investigates the connection between the branching Markov processes constructed over some given Markov processes and a non-linear equation close to Boltzmann's equation. A special class of excessive measures for the branching Markov process is described and studied, as well as the corresponding dual processes
Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see
618,
1011Keywords: Boltzmann equation,
Branching processesNature: Original Retrieve article from Numdam
IX: 28, 494-494, LNM 465 (1975)
DELLACHERIE, Claude
Correction à ``Intégrales stochastiques par rapport...'' (
General theory of processes)
This paper completes a gap in the simple proof of the previsible representation property of the Wiener process, given by Dellacherie
805Comment: Another way of filling this gap is given by Ruiz de Chavez
1821. The same gap for the Poisson process is corrected in
2002Keywords: Previsible representationNature: Original Retrieve article from Numdam
IX: 29, 495-495, LNM 465 (1975)
DELLACHERIE, Claude
Une propriété des ensembles semi-polaires (
Markov processes)
It is shown that semi-polar sets are exactly those which have potential 0 for all continuous additive functionals (or for all time-changed processes)
Keywords: Semi-polar setsNature: Original Retrieve article from Numdam
IX: 30, 496-514, LNM 465 (1975)
SHARPE, Michael J.
Homogeneous extensions of random measures (
Markov processes)
Homogeneous random measures are the appropriate definition of additive functionals which may explode. The problem discussed here is the extension of such a measure given up to a terminal time into a measure defined up to the lifetime
Comment: The subject is taken over in a systematic way in Sharpe,
General Theory of Markov processes, Academic Press 1988
Keywords: Homogeneous random measures,
Terminal times,
SubprocessesNature: Original Retrieve article from Numdam
IX: 31, 515-517, LNM 465 (1975)
HEATH, David C.
Skorohod stopping in discrete time (
Markov processes,
Potential theory)
Using ideas of Mokobodzki, it is shown how the imbedding of a measure $\mu_1$ in the discrete Markov process with initial measure $\mu_0$ can be achieved by a random mixture of hitting times
Comment: This is a potential theoretic version of the original construction of Skorohod. This paper is better read in conjunction with Heath
811. A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
IX: 32, 518-521, LNM 465 (1975)
MAISONNEUVE, Bernard;
MEYER, Paul-André
Ensembles aléatoires markoviens homogènes. Mise au point et compléments (
Markov processes)
This paper corrects or simplifies many details in the long paper
713 by the same authors
Comment: See also the next paper
933Keywords: Regenerative systems,
Last-exit decompositions,
ExcursionsNature: Original Retrieve article from Numdam
IX: 33, 522-529, LNM 465 (1975)
MAISONNEUVE, Bernard
Le comportement de dernière sortie (
Markov processes)
This paper contains improvements to the paper
813 by Maisonneuve-Meyer, whose results are briefly recalled. Incursion processes and Lévy systems are altogether avoided, last-exist decompositions are derived, and the strong Markov property of the analogue of the age process in renewal theory is proved, as well as a non-homogeneous Markov property for some processes starting at last-exit times. The extension of these results to abstractly defined regenerative systems is mentioned
Comment: More detailed versions of these results appear in Maisonneuve,
Ann. Prob.,
3, 1975,
Z. für W-theorie, 80, 1989, and in Chapter XX of Dellacherie-Maisonneuve-Meyer,
Probabilités et Potentiel, Hermann 1992
Keywords: Regenerative systems,
Last-exit decompositions,
ExcursionsNature: Original Retrieve article from Numdam
IX: 35, 534-554, LNM 465 (1975)
EL KAROUI, Nicole
Processus de réflexion dans ${\bf R}^n$ (
Diffusion theory)
In the line of the seminar on diffusions
419 this talk presents the theory of diffusions in a half space with continuous coefficients and a boundary condition on the boundary hyperplane involving a reflexion part, but more general than the pure reflexion case considered by Stroock-Varadhan (
Comm. Pure Appl. Math.,
24, 1971). The point of view is that of martingale problems
Comment: This talk is a late publication of work done by the author in 1971
Keywords: Boundary reflection,
Local timesNature: Original Retrieve article from Numdam
IX: 36, 555-555, LNM 465 (1975)
MEYER, Paul-André
Une remarque sur les processus de Markov (
Markov processes)
It is shown that, under a fixed measure $
P^{\mu}$, the optional processes and times relative to the uncompleted filtrations $({\cal F}_{t+}^{\circ})$ and $({\cal F}_{t}^{\circ})$ are undistinguishable from each other
Comment: No applications are known
Keywords: Stopping timesNature: Original Retrieve article from Numdam
IX: 37, 556-564, LNM 465 (1975)
MEYER, Paul-André
Retour aux retournements (
Markov processes,
General theory of processes)
The first part of the talk is devoted to an important correction to the theorem on p.285 of
814 (Azéma's theory of cooptional and coprevisible sets and its application to Markov processes): the definition of left additive functionals should allow an a.s. explosion at time 0 for initial points belonging to a polar set. The second part belongs to the general theory of processes: if the index set is not the right half-line as usual but the left half-line, the section and projection theorems must be modified in a not quite trivial way
Comment: See Chapter XXVIII of Dellacherie-Maisonneuve-Meyer
Probabilités et potentiel Keywords: Time reversal,
Cooptional processes,
Coprevisible processes,
Homogeneous processesNature: Exposition,
Original additions Retrieve article from Numdam
IX: 38, 565-588, LNM 465 (1975)
WALDENFELS, Wilhelm von
Interval partitions and pair interactions (
Miscellanea)
To be completed
Comment: To be completed
Nature: Original Retrieve article from Numdam
X: 01, 1-18, LNM 511 (1976)
BRÉMAUD, Pierre
La méthode des semi-martingales en filtrage quand l'observation est un processus ponctuel marqué (
Martingale theory,
Point processes)
This paper discusses martingale methods (as developed by Jacod,
Z. für W-theorie, 31, 1975) in the filtering theory of point processes
Comment: The author has greatly developed this topic in his book
Poisson Processes and Queues, Springer 1981
Keywords: Point processes,
Previsible representation,
Filtering theoryNature: Original Retrieve article from Numdam
X: 02, 19-23, LNM 511 (1976)
CHACON, Rafael V.;
WALSH, John B.
One-dimensional potential imbedding (
Brownian motion)
The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool
Comment: The construction is related to that of Dubins (see
516). In this volume
1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
X: 03, 24-39, LNM 511 (1976)
JACOD, Jean;
MÉMIN, Jean
Un théorème de représentation des martingales pour les ensembles régénératifs (
Martingale theory,
Markov processes,
Stochastic calculus)
The natural filtration of a regenerative set $M$ is that of the corresponding ``age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\mu-\nu$ has the previsible representation property in the natural filtration
Comment: Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in
1932. In the case of the set of zeros of Brownian motion, the martingale considered here is the second ``Azéma's martingale'' (not the well known one which has the chaotic representation property)
Keywords: Regenerative sets,
Renewal theory,
Stochastic integrals,
Previsible representationNature: Original Retrieve article from Numdam
X: 04, 40-43, LNM 511 (1976)
KAZAMAKI, Norihiko
A simple remark on the conditioned square functions for martingale transforms (
Martingale theory)
This is a problem of discrete martingale theory, giving inequalities between the conditioned square funtions (discrete angle brackets) of martingale transforms of two martingales related through a change of time
Comment: The author has published a paper on a related subject in
Tôhoku Math. J.,
28, 1976
Keywords: Angle bracket,
InequalitiesNature: Original Retrieve article from Numdam
X: 05, 44-77, LNM 511 (1976)
KUNITA, Hiroshi
Absolute continuity for Markov processes (
Markov processes)
This paper is devoted to a ``progressive'' Lebesgue decomposition of the laws of a Markov process with respect to a second one in the same filtration, and the structure of the corresponding density. The two processes are assumed to be Hunt processes, and for part of the paper satisfy Hunt's hypothesis (K) (all excessive functions are regular, or semi-polar sets are polar). The topics discussed are the following: Lévy systems and the relation between the Lévy systems of a process and of its transform by a multiplicative functional; structure of exact perfect terminal times, which are shown to be hitting times of sets in space-time, by the process $(X_{t-},X_t)$ (a version of a result of Walsh-Weil,
Ann. Sci. ENS, 5, 1972); the ``Lebesgue decomposition'' of a Markov process with respect to another, and the fact that if absolute continuity holds on the germ field it also holds up to some maximal terminal time; a condition for this terminal time to be equal to the lifetime, under hypothesis (K)
Comment: The pasting together of the Lebesgue decompositions of a probability measure with respect to another one, on the $\sigma$-fields of a given filtration, is called the
Kunita decomposition, and is not restricted to Markov processes. For the general case, see Yoeurp, in LN
1118,
Grossissements de filtrations, 1985
Keywords: Absolute continuity of laws,
Hunt processes,
Terminal times,
Kunita decompositionNature: Original Retrieve article from Numdam
X: 06, 78-85, LNM 511 (1976)
MANDREKAR, Vidyadhar
Germ-field Markov property for multiparameter processes (
Miscellanea)
The paper studies the relations between several Markov properties of a process indexed by an open set of $
R^n$
Comment: To be completed
Keywords: Several parameter Brownian motions,
Several parameter processesNature: Original Retrieve article from Numdam
X: 07, 86-103, LNM 511 (1976)
MEYER, Paul-André
La théorie de la prédiction de F. Knight (
General theory of processes)
This paper is devoted to the work of Knight,
Ann. Prob. 3, 1975, the main idea of which is to associate with every reasonable process $(X_t)$ another process, taking values in a space of probability measures, and whose value at time $t$ is a conditional distribution of the future of $X$ after $t$ given its past before $t$. It is shown that the prediction process contains essentially the same information as the original process (which can be recovered from it), and that it is a time-homogeneous Markov process
Comment: The results are related to those of Schwartz (presented in
722), the main difference being that the future is predicted instead of the whole path. Knight has devoted to this subject the
Essays on the Prediction Process, Hayward Inst. of Math. Stat., 1981, and a book,
Foundations of the Prediction Process, Oxford Science Publ. 1992
Keywords: Prediction theoryNature: Exposition,
Original additions Retrieve article from Numdam
X: 08, 104-117, LNM 511 (1976)
MEYER, Paul-André;
YOR, Marc
Sur la théorie de la prédiction, et le problème de décomposition des tribus ${\cal F}^{\circ}_{t+}$ (
General theory of processes)
This paper contains another version of Knight's theory (preceding paper
1007) for cadlag process instead of measurable processes. These results then are applied to the pathology of germ fields: a natural measurability conjecture does not hold, and an example is given of a process $X_t$ such that its natural $\sigma$-field ${\cal F}_{1+}$ is not generated by ${\cal F}_{1}$ and the germ-field at $0$ of the process $(X_{1+s})$
Comment: On the pathology of germ fields, see H. von Weizsäcker,
Ann. Inst. Henri Poincaré, 19, 1983
Keywords: Prediction theory,
Germ fieldsNature: Original Retrieve article from Numdam
X: 09, 118-124, LNM 511 (1976)
MEYER, Paul-André
Generation of $\sigma$-fields by step processes (
General theory of processes)
On a Blackwell measurable space, let ${\cal F}_t$ be a right continuous filtration, such that for any stopping time $T$ the $\sigma$-field ${\cal F}_T$ is countably generated. Then (discarding possibly one single null set), this filtration is the natural filtration of a right-continuous step process
Comment: This answers a question of Knight,
Ann. Math. Stat.,
43, 1972
Keywords: Point processesNature: Original Retrieve article from Numdam
X: 10, 125-183, LNM 511 (1976)
MEYER, Paul-André
Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (
Applications of martingale theory,
Markov processes)
This long paper consists of four talks, suggested by E.M.~Stein's book
Topics in Harmonic Analysis related to the Littlewood-Paley theory, Princeton 1970. The classical Littlewood-Paley theory shows that the $L^p$ norm ($1<p<\infty$) of a function $f$ on $
R^n$ is equivalent to that of several kinds of non-linear functionals of $f$ called Littlewood-Paley functions, which are square roots of quadratic expressions involving the harmonic extension of $f$ to the half-space $
R^n\times
R_+$, and its derivatives. Using these equivalences, it is easy to prove that the Riesz transforms are bounded in~$L^p$. The classical theory is given a probabilistic interpretation, the L-P functions appearing as conditional expectations of functionals of a Brownian motion on the half-space, given its final position on the limit hyperplane, and then the L-P inequalities follow from the Burkholder inequalities of martingale theory. The original L-P theory concerned the unit disk; Stein had extended it to $
R^n$ and had started extending it to symmetric semigroups. Here a new tool is introduced, the squared-field operator (carré du champ) introduced by J.P.~Roth (
CRAS Paris, 278A, 1974, p.1103) in potential theory and by Kunita (
Nagoya M. J.,
36, 1969) in probability. This paper consists of 4 talks, and in the last one theorems 1' and 3 are false
Comment: This paper was rediscovered by Varopoulos (
J. Funct. Anal.,
38, 1980), and was then rewritten by Meyer in
1510 in a simpler form. Its main application has been to the Ornstein-Uhlenbeck semigroup in
1816. It has been superseded by the theory of $\Gamma_2$ due to Bakry
1910, see also Bakry-Émery
1912, and Meyer
1908 reporting on Cowling's extension of Stein's work. An erratum is given in
1253Keywords: Littlewood-Paley theory,
Riesz transforms,
Brownian motion,
Inequalities,
Harmonic functions,
Singular integrals,
Carré du champ,
Infinitesimal generators,
Semigroup theoryNature: Original Retrieve article from Numdam
X: 11, 184-193, LNM 511 (1976)
NAGASAWA, Masao
A probabilistic approach to non-linear Dirichlet problem (
Markov processes)
The theory of branching Markov processes in continuous time developed in particular by Ikeda-Nagasawa-Watanabe (
J. Math. Kyoto Univ.,
8, 1968 and
9, 1969) and Nagasawa (
Kodai Math. Sem. Rep. 20, 1968) leads to the probabilistic solution of a non-linear Dirichlet problem
Comment: For other contributions by the same author devoted to the relation between branching process and non-linear equations, see
618,
926Keywords: Branching processes,
Dirichlet problemNature: Original Retrieve article from Numdam
X: 12, 194-208, LNM 511 (1976)
ROST, Hermann
Skorohod stopping times of minimal variance (
Markov processes)
Root's (
Ann. Math. Stat.,
40, 1969) solution of the Shorohod imbedding problem for Brownian motion uses the hitting time of a barrier in space-time. Here Root's construction is extended to general Markov processes, an optimality property of Root's stopping times is proved, as well as the uniqueness of such stopping times
Comment: For previous work of the author on Skorohod's imbedding see
Ann. M. Stat. 40, 1969 and
Invent. Math. 14, 1971, and in this Seminar
523,
613,
806. A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
X: 13, 209-215, LNM 511 (1976)
SEKIGUCHI, Takesi
On the Krickeberg decomposition of continuous martingales (
Martingale theory)
The problem investigated is whether the two positive martingales occurring in the Krickeberg decomposition of a $L^1$-bounded continuous martingale of a filtration $({\cal F}_t)$ are themselves continuous. It is shown that the answer is yes only under very stringent conditions: there exists a sub-filtration $({\cal G}_t)$ such that 1) all ${\cal G}$-martingales are continuous 2) the continuous ${\cal F}$-martingales are exactly the ${\cal G}$-martingales
Comment: For related work of the author see
Tôhoku Math. J. 28, 1976
Keywords: Continuous martingales,
Krickeberg decompositionNature: Original Retrieve article from Numdam
X: 14, 216-234, LNM 511 (1976)
WILLIAMS, David
The Q-matrix problem (
Markov processes)
This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched
Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams
Diffusions, Markov Processes and Martingales, vol. 1 (second edition), Wiley 1994. See also
1024Keywords: Markov chains,
Ray compactification,
Local times,
ExcursionsNature: Original Retrieve article from Numdam
X: 15, 235-239, LNM 511 (1976)
WILLIAMS, David
On a stopped Brownian motion formula of H.M.~Taylor (
Brownian motion)
This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given
Comment: For the original proof of Taylor see
Ann. Prob. 3, 1975. For modern references, we should ask Yor
Keywords: Stopping times,
Local times,
Ray-Knight theorems,
Cameron-Martin formulaNature: Original Retrieve article from Numdam
X: 16, 240-244, LNM 511 (1976)
YAMADA, Toshio
On the uniqueness of solutions of stochastic differential equations with reflecting barrier conditions (
Stochastic calculus,
Diffusion theory)
A stochastic differential equation is considered on the positive half-line, driven by Brownian motion, with time-dependent coefficients and a reflecting barrier condition at $0$ (Skorohod style). Skorohod proved pathwise uniqueness under Lipschitz condition, and this is extended here to moduli of continuity satisfying integral conditions
Comment: This extends to the reflecting barrier case the now classical result in the ``free'' case due to Yamada-Watanabe,
J. Math. Kyoto Univ.,
11, 1971. Many of these theorems have now simpler proofs using local times, in the spirit of Revuz-Yor,
Continuous Martingales and Brownian Motion, Chapter IX
Keywords: Stochastic differential equations,
Boundary reflectionNature: Original Retrieve article from Numdam
X: 17, 245-400, LNM 511 (1976)
MEYER, Paul-André
Un cours sur les intégrales stochastiques (6 chapters) (
Stochastic calculus,
Martingale theory,
General theory of processes)
This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$
Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also
1131. Now the material can be found in many books
Keywords: Increasing processes,
Stable subpaces,
Angle bracket,
Square bracket,
Stochastic integrals,
Optional stochastic integrals,
Previsible representation,
Change of variable formula,
Semimartingales,
Stochastic exponentials,
Multiplicative decomposition,
Fefferman inequality,
Davis inequality,
Stratonovich integrals,
Burkholder inequalities,
$BMO$,
Multiple stochastic integrals,
Girsanov's theoremNature: Exposition,
Original additions Retrieve article from Numdam
X: 18, 401-413, LNM 511 (1976)
PRATELLI, Maurizio
Sur certains espaces de martingales de carré intégrable (
Martingale theory)
The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$
Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli,
1404. An error is corrected in
1250Keywords: Inequalities,
Angle bracket,
$BMO$Nature: Original Retrieve article from Numdam
X: 19, 414-421, LNM 511 (1976)
PRATELLI, Maurizio
Espaces fortement stables de martingales de carré intégrable (
Martingale theory,
Stochastic calculus)
This paper studies closed subspaces of the Hilbert space of square integrable martingales which are stable under optional stochastic integration (see
1018)
Keywords: Stable subpaces,
Square integrable martingales,
Stochastic integrals,
Optional stochastic integralsNature: Original Retrieve article from Numdam
X: 20, 422-431, LNM 511 (1976)
YAN, Jia-An;
YOEURP, Chantha
Représentation des martingales comme intégrales stochastiques des processus optionnels (
Martingale theory,
Stochastic calculus)
An attempt to build a theory similar to the previsible representation property with respect to a basic local martingale, but using the optional stochastic integral instead of the standard one
Comment: Apparently this ``optional representation property'' has not been used since
Keywords: Optional stochastic integralsNature: Original Retrieve article from Numdam
X: 21, 432-480, LNM 511 (1976)
YOEURP, Chantha
Décomposition des martingales locales et formules exponentielles (
Martingale theory,
Stochastic calculus)
It is shown that local martingales can be decomposed uniquely into three pieces, a continuous part and two purely discontinuous pieces, one with accessible jumps, and one with totally inaccessible jumps. Two beautiful lemmas say that a purely discontinuous local martingale whose jumps are summable is a finite variation process, and if it has accessible jumps, then it is the sum of its jumps without compensation. Conditions are given for the existence of the angle bracket of two local martingales which are not locally square integrable. Lemma 2.3 is the lemma often quoted as ``Yoeurp's Lemma'': given a local martingale $M$ and a previsible process of finite variation $A$, $[M,A]$ is a local martingale. The definition of a local martingale on an open interval $[0,T[$ is given when $T$ is previsible, and the behaviour of local martingales under changes of laws (Girsanov's theorem) is studied in a set up where the positive martingale defining the mutual density is replaced by a local martingale. The existence and uniqueness of solutions of the equation $Z_t=1+\int_0^t\tilde Z_s dX_s$, where $X$ is a given special semimartingale of decomposition $M+A$, and $\widetilde Z$ is the previsible projection of the unknown special semimartingale $Z$, is proved under an assumption that the jumps $ėlta A_t$ do not assume the value $1$. Then this ``exponential'' is used to study the multiplicative decomposition of a positive supermartingale in full generality
Comment: The problems in this paper have some relation with Kunita
1005 (in a Markovian set up), and are further studied by Yoeurp in LN
1118,
Grossissements de filtrations, 1985. The subject of multiplicative decompositions of positive submartingales is much more difficult since they may vanish. For a simple case see in this volume Yoeurp-Meyer
1023. The general case is due to Azéma (
Z. für W-theorie, 45, 1978, presented in
1321) See also
1622Keywords: Stochastic exponentials,
Multiplicative decomposition,
Angle bracket,
Girsanov's theorem,
Föllmer measuresNature: Original Retrieve article from Numdam
X: 22, 481-500, LNM 511 (1976)
YOR, Marc
Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles (
Martingale theory,
Stochastic calculus)
This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula ${\cal E}(X)\,{\cal E}(Y)={\cal E}(X+Y+[X,Y])$ where ${\cal E}$ denotes the usual exponential of semimartingales. Also, the s.d.e. $Z_t=1+\int_0^t Z_sdX_s$ is solved, where $X$ is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of ``exponentials'' is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita-Watanabe in a Markovian set-up
Keywords: Optional stochastic integrals,
Stochastic exponentials,
Lévy systemsNature: Original Retrieve article from Numdam
X: 23, 501-504, LNM 511 (1976)
MEYER, Paul-André;
YOEURP, Chantha
Sur la décomposition multiplicative des sousmartingales positives (
Martingale theory)
This paper expands part of Yoeurp's paper
1021, to cover the decomposition of positive submartingales instead supermartingales, assuming that the process never vanishes. A corollary is that every positive (not necessarily strictly so) submartingale $X_t$ is the optional projection of an increasing process $C_t$, non-adapted, such that $0\leq C_t\leq X_{\infty}$
Comment: See the comments on
1021 for the general case. The latter result is related to Meyer
817. For a related paper, see
1203. Further study in
1620Keywords: Multiplicative decompositionNature: Original Retrieve article from Numdam
X: 24, 505-520, LNM 511 (1976)
WILLIAMS, David
The Q-matrix problem 2: Kolmogorov backward equations (
Markov processes)
This is an addition to
1014, the problem being now of constructing a chain whose transition probabilities satisfy the Kolmogorov backward equations, as defined in a precise way in the paper. A different construction is required
Keywords: Markov chainsNature: Original Retrieve article from Numdam
X: 25, 521-531, LNM 511 (1976)
BENVENISTE, Albert
Séparabilité optionnelle, d'après Doob (
General theory of processes)
A real valued function $f(t)$ admits a countable set $D$ as a separating set if the graph of $f$ is contained in the closure of its restriction to $D$. Doob's well known theorem asserts that every process $X$ has a modification all sample functions of which admit a common separating set $D$ (deterministic). It is shown that if $D$ is allowed to consist of (the values of) countably many stopping times, then every optional process is separable without modification. Applications are given
Comment: Doob's original paper appeared in
Ann. Inst. Fourier, 25, 1975. See also
1105Keywords: Optional processes,
Separability,
Section theoremsNature: Exposition,
Original additions Retrieve article from Numdam
X: 26, 532-535, LNM 511 (1976)
NAGASAWA, Masao
Note on pasting of two Markov processes (
Markov processes)
The pasting or piecing out theorem says roughly that two Markov processes taking values in two open sets and agreeing up to the first exit time of their intersection can be extended into a single Markov process taking values in their union. The word ``roughly'' replaces a precise definition, necessary in particular to handle jumps. Though the result is intuitively obvious, its proof is surprisingly messy. It is due to Courrège-Priouret,
Publ. Inst. Stat. Univ. Paris, 14, 1965. Here it is reduced to a ``revival theorem'' of Ikeda-Nagasawa-Watanabe,
J. Math. Kyoto Univ.,
8, 1968
Comment: The piecing out theorem is also reduced to a revival theorem in Meyer,
Ann. Inst. Fourier, 25,1975
Keywords: Piecing-out theoremNature: Original Retrieve article from Numdam
X: 27, 536-539, LNM 511 (1976)
KAZAMAKI, Norihiko
A characterization of $BMO$ martingales (
Martingale theory)
A $L^2$ bounded continuous martingale belongs to $BMO$ if and only if its stochastic exponential satisfies some (Muckenhoupt) condition $A_p$ for $p>1$
Comment: For an extension to non-continuous martingales, see
1125. For a recent survey see the monograph of Kazamaki on exponential martingales and $BMO$, LN
1579, 1994
Keywords: $BMO$Nature: Original Retrieve article from Numdam
X: 28, 540-543, LNM 511 (1976)
MOKOBODZKI, Gabriel
Démonstration élémentaire d'un théorème de Novikov (
Descriptive set theory)
Novikov's theorem asserts that any sequence of analytic subsets of a compact metric space with empty intersection can be enclosed in a sequence of Borel sets with empty intersection. This result has important consequences in descriptive set theory (see Dellacherie
915). A fairly simple proof of this theorem is given, which relates it to the first separation theorem (rather than the second separation theorem as it used to be)
Comment: Dellacherie in this volume (
1032) further simplifies the proof. For a presentation in book form, see Dellacherie-Meyer,
Probabilités et Potentiel C, chapter XI
9Keywords: Analytic setsNature: Original Retrieve article from Numdam
X: 29, 544-544, LNM 511 (1976)
DELLACHERIE, Claude
Correction à des exposés de 1973/74 (
Descriptive set theory)
Corrections to
915 and
918Keywords: Analytic sets,
Semi-polar sets,
Suslin spacesNature: Original Retrieve article from Numdam
X: 30, 545-577, LNM 511 (1976)
DELLACHERIE, Claude
Sur la construction de noyaux boréliens (
Measure theory)
This answers questions of Getoor
923 and Meyer
924 on the regularization of a pseudo-kernel relative to a family ${\cal N}$ of negligible sets into a Borel kernel. The problem is reduced to a simpler one, whether a non-negligible set $A$ contains a non-negligible Borel set, which itself is answered in the affirmative if 1) The underlying space is compact metric, 2) $A$ is coanalytic, 3) ${\cal N}$ consists of all sets negligible for all measures of an analytic family. The proof uses general methods, of independent interest
Comment: For a presentation in book form, see Dellacherie-Meyer,
Probabilités et Potentiel C, chapter XI
41. The hypothesis that the space is compact is sometimes troublesome for the applications
Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
X: 32, 579-593, LNM 511 (1976)
DELLACHERIE, Claude
Compléments aux exposés sur les ensembles analytiques (
Descriptive set theory)
A new proof of Novikov's theorem (see
1028 and the corresponding comments) is given in the form of a Choquet theorem for multicapacities (with infinitely many arguments). Another (unrelated) result is a complement to
919 and
920, which study the space of stopping times. The language of stopping times is used to prove a deep section theorem due to Kondo
Keywords: Analytic sets,
Section theorems,
CapacitiesNature: Original Retrieve article from Numdam
XI: 01, 1-20, LNM 581 (1977)
AVANISSIAN, Vazgain
Fonctions harmoniques d'ordre infini et l'harmonicité réelle liée à l'opérateur laplacien itéré (
Potential theory,
Miscellanea)
This paper studies two classes of functions in (an open set of) $
R^n$, $n\ge1$: 1) Harmonic functions of infinite order (see Avanissian and Fernique,
Ann. Inst. Fourier, 18-2, 1968), which are $C^\infty$ functions satisfying a growth condition on their iterated laplacians, and are shown to be real analytic. 2) Infinitely differentiable functions (or distributions) similar to completely monotonic functions on the line, i.e., whose iterated laplacians are alternatively positive and negative (they were introduced by Lelong). Among the results is the fact that the second class is included in the first
Keywords: Harmonic functions,
Real analytic functions,
Completely monotonic functionsNature: Original Retrieve article from Numdam
XI: 02, 21-26, LNM 581 (1977)
BENVENISTE, Albert
Application d'un théorème de G. Mokobodzki à la théorie des flots (
Ergodic theory,
General theory of processes)
The purpose of this paper is to extend to the theory of filtered flows (for which see
901 and
902) the dual version of the general theory of processes due to Azéma (for which see
814 and
937), in particular the association with any measurable process of suitable projections which are homogeneous processes. An important difference here is the fact that the time set is the whole line. Here the class of measurable processes which can be projected is reduced to a (not very explicit) class, and a commutation theorem similar to Azéma's is proved. The proof uses the technique of
medial limits due to Mokobodzki (see
719), which in fact was developed precisely at the author's request to solve this problem
Keywords: Filtered flows,
Stationary processes,
Projection theorems,
Medial limitsNature: Original Retrieve article from Numdam
XI: 04, 34-46, LNM 581 (1977)
DELLACHERIE, Claude
Les dérivations en théorie descriptive des ensembles et le théorème de la borne (
Descriptive set theory)
At the root of set theory lies Cantor's definition of the ``derived set'' $\delta A$ of a closed set $A$, i.e., the set of its non-isolated points, with the help of which Cantor proved that a closed set can be decomposed into a perfect set and a countable set. One may define the index $j(A)$ to be the smallest ordinal $\alpha$ such that $\delta^\alpha A=\emptyset$, or $\omega_1$ if there is no such ordinal. Considering the set $F$ of all closed sets as a (Polish) topological space, ordered by inclusion, $\delta$ as an increasing mapping from $F$ such that $\delta A\subset A$, let $D$ be the set of all $A$ such that $j(A)<\omega_1$ (thus, the set of all countable closed sets). Then $D$ is coanalytic and non-Borel, while the index is bounded by a countable ordinal on every analytic subset of $D$. These powerful results are stated abstractly and proved under very general conditions. Several examples are given
Comment: See a correction in
1241, and several examples in Hillard
1242. The whole subject has been exposed anew in Chapter~XXIV of Dellacherie-Meyer,
Probabilités et PotentielKeywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Exposition,
Original additions Retrieve article from Numdam
XI: 05, 47-50, LNM 581 (1977)
DELLACHERIE, Claude
Deux remarques sur la séparabilité optionnelle (
General theory of processes)
Optional separability was defined by Doob,
Ann. Inst. Fourier, 25, 1975. See also Benveniste,
1025. The main remark in this paper is the following: given any optional set $H$ with countable dense sections, there exists a continuous change of time $(T_t)$ indexed by $[0,1[$ such that $H$ is the union of all graphs $T_t$ for $t$ dyadic. Thus Doob's theorem amounts to the fact that every optional process becomes separable in the ordinary sense once a suitable continuous change of time has been performed
Keywords: Optional processes,
Separability,
Changes of timeNature: Original Retrieve article from Numdam
XI: 06, 51-58, LNM 581 (1977)
DUDLEY, Richard M.;
GUTMANN, Sam
Stopping times with given laws (
General theory of processes)
Given a filtration ${\cal A}_t$ such that for all $t>0$ the law $P$ restricted to ${\cal A}_t$ is non-atomic, then for every law $\mu$ on the half-line there exists a stopping time with law $\mu$. The proof uses an interesting measure theoretic lemma on decreasing sequences of non-atomic $\sigma$-fields
Comment: It follows that the Brownian filtration contains a process whose law is that of a Poisson process (though of course it is not a Poisson process
of the Brownian filtration )
Keywords: StoppingNature: Original Retrieve article from Numdam
XI: 07, 59-64, LNM 581 (1977)
HOROWITZ, Joseph
Une remarque sur les bimesures (
Measure theory)
A bimeasure is a function $\beta(A,B)$ of two set variables, which is a measure in each variable when the other is kept fixed. It is important to have conditions under which a bimeasure ``is'' a measure, i.e., is of the form $\mu(A\times B)$ for some measure $\mu$ on the product space. This is known to be true for positive bimeasures ( Kingman,
Pacific J. of Math., 21, 1967, see also
315). Here a condition of bounded variation is given, which implies that a bimeasure is a difference of two positive bimeasures, and therefore is a measure
Comment: Signed bimeasures which are not measures occur naturally, see for instance Bakry
1742, and Émery-Stricker on Gaussian semimartingales
Keywords: BimeasuresNature: Original Retrieve article from Numdam
XI: 08, 65-78, LNM 581 (1977)
EL KAROUI, Nicole;
MEYER, Paul-André
Les changements de temps en théorie générale des processus (
General theory of processes)
Given a filtration $({\cal F}_t)$ and a continuous adapted increasing process $(C_t)$, consider its right inverse $(j_t)$ and left inverse $(i_t)$, and the time-changed filtration $\overline{\cal F}_t={\cal F}_{j_t}$. The problem is to study the relation between optional/previsible processes of the time-changed filtration and time-changed optional/previsible processes of the original filtration, to see how the projections or dual projections are related, etc. The results are satisfactory, and require a lot of care
Comment: This paper was originally an exposition by the second author of an unpublished paper of the first author, and many ``I''s remained in spite of the final joint autorship. See the next paper
1109 for the discontinuous case
Keywords: Changes of timeNature: Original Retrieve article from Numdam
XI: 09, 79-108, LNM 581 (1977)
EL KAROUI, Nicole;
WEIDENFELD, Gérard
Théorie générale et changement de temps (
General theory of processes)
The results of the preceding paper
1108 are extended to arbitrary changes of times, i.e., without the continuity assumption on the increasing process. They require even more care
Comment: Unfortunately, the material presentation of this paper is rather poor. For related results, see
1333Keywords: Changes of timeNature: Original Retrieve article from Numdam
XI: 12, 132-195, LNM 581 (1977)
MEYER, Paul-André
Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (
Potential theory,
Applications of martingale theory)
This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(
R^n)$ is $BMO$, using methods adapted from the probabilistic Littlewood-Paley theory (of which this is a kind of limiting case). Some details of the proof are interesting in their own right
Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see
1818. The reasoning around (3.1) p.178 needs to be corrected
Keywords: Harmonic functions,
Hardy spaces,
Poisson kernel,
Carleson measures,
$BMO$,
Riesz transformsNature: Exposition,
Original additions Retrieve article from Numdam
XI: 13, 196-256, LNM 581 (1977)
WEBER, Michel
Classes uniformes de processus gaussiens stationnaires (
Gaussian processes)
To be completed
Comment: See the long and interesting review by Berman in
Math. Reviews, 56,
13343Nature: Exposition,
Original additions Retrieve article from Numdam
XI: 14, 257-297, LNM 581 (1977)
YOR, Marc
Sur les théories du filtrage et de la prédiction (
General theory of processes,
Markov processes)
To be completed
Comment: MR 57, 10801
Keywords: Filtering theory,
Prediction theoryNature: Original Retrieve article from Numdam
XI: 15, 298-302, LNM 581 (1977)
ZANZOTTO, Pio Andrea
Sur l'existence d'un noyau induisant un opérateur sous markovien donné (
Measure theory)
The problem is whether a positive, norm-decreasing operator $L^\infty(\mu)\rightarrow L^\infty(\lambda)$ (of classes, not functions) is induced by a submarkov kernel. No ``countable additivity'' condition is assumed, but completeness of $\lambda$ and tightness of $\mu$
Comment: See
923,
924,
1030Keywords: Pseudo-kernels,
RegularizationNature: Original Retrieve article from Numdam
XI: 16, 303-323, LNM 581 (1977)
BERNARD, Alain;
MAISONNEUVE, Bernard
Décomposition atomique de martingales de la classe $H^1$ (
Martingale theory)
Atomic decompositions have been used with great success in the analytical theory of Hardy spaces, in particular by Coifman (
Studia Math. 51, 1974). An atomic decomposition of a Banach space consists in finding simple elements (called atoms) in its unit ball, such that every element is a linear combination of atoms $\sum_n \lambda_n a_n$ with $\sum_n \|\lambda_n\|<\infty$, the infimum of this sum defining the norm or an equivalent one. Here an atomic decomposition is given for $H^1$ spaces of martingales in continuous time (defined by their maximal function). Atoms are of two kinds: the first kind consists of martingales bounded uniformly by a constant $c$ and supported by an interval $[T,\infty[$ such that $P\{T<\infty\}\le 1/c$. These atoms do not generate the whole space $H^1$ in general, though they do in a few interesting cases (if all martingales are continuous, or in the discrete dyadic case). To generate the whole space it is sufficient to add martingales of integrable variation (those whose total variation has an $L^1$ norm smaller than $1$ constitute the second kind of atoms). This approach leads to a proof of the $H^1$-$BMO$ duality and the Davis inequality
Comment: See also
1117Keywords: Atomic decompositions,
$H^1$ space,
$BMO$Nature: Original Retrieve article from Numdam
XI: 17, 324-326, LNM 581 (1977)
BERNARD, Alain
Complément à l'exposé précédent (
Martingale theory)
This paper is a sequel to
1116, which it completes in two ways: it makes it independent of a previous proof of the Fefferman inequality, which is now proved directly, and it exhibits atoms of the first kind appropriate to the quadratic norm of $H^1$
Keywords: Atomic decompositions,
$H^1$ space,
$BMO$Nature: Original Retrieve article from Numdam
XI: 18, 327-339, LNM 581 (1977)
CAIROLI, Renzo;
WALSH, John B.
Prolongement de processus holomorphes. Cas ``carré intégrable'' (
Several parameter processes)
This paper concerns a class of two-parameter (real) processes adapted to the filtration of the Brownian sheet, and called holomorphic in the seminal paper of the authors in
Acta Math. 4, 1975. These processes have stochastic integral representations along (increasing) paths, with a common kernel called their derivative. Under an integrability restriction, a process holomorphic in a region of the plane is shown to be extendable as a holomorphic process to a larger region of a canonical shape (intersection of a rectangle and a disk centered at the origin)
Comment: See also
1119Keywords: Holomorphic processes,
Brownian sheetNature: Original Retrieve article from Numdam
XI: 19, 340-348, LNM 581 (1977)
CAIROLI, Renzo;
WALSH, John B.
Some examples of holomorphic processes (
Several parameter processes)
This is a sequel to the preceding paper
1118. It also extends the definition to processes defined on a random domain
Comment: See the author's paper in
Ann. Prob. 5, 1971 for additional results
Keywords: Holomorphic processes,
Brownian sheetNature: Original Retrieve article from Numdam
XI: 20, 349-355, LNM 581 (1977)
CAIROLI, Renzo;
WALSH, John B.
On changing time (
Several parameter processes)
The analogue of the well-known result that any continuous martingale can be time changed into a Brownian motion using its own quadratic variation process is answered negatively for two-parameter martingales (even strong ones) in the filtration of the Brownian sheet
Keywords: Brownian sheetNature: Original Retrieve article from Numdam
XI: 21, 356-361, LNM 581 (1977)
CHOU, Ching Sung
Le processus des sauts d'une martingale locale (
Martingale theory)
Simple necessary and sufficient conditions are given on an optional process $\sigma_t$ (different from $0$ only at countably many stopping times) so that it is the process of jumps $ėlta M_t$ of some local martingale $M$
Comment: The same result is proved independently by D. Lépingle in this volume, see
1129. For an application see
1308, and for another approach
1335Keywords: Local martingales,
JumpsNature: Original Retrieve article from Numdam
XI: 22, 362-364, LNM 581 (1977)
DELLACHERIE, Claude
Sur la régularisation des surmartingales (
Martingale theory)
It is shown that any supermartingale has a version which is strong, i.e., which is optional and satisfies the supermartingale inequality at bounded stopping times, even if the filtration does not satisfy the usual conditions (and under the usual conditions, without assuming the expectation to be right-continuous)
Comment: See
1524Keywords: General filtrations,
Strong supermartingalesNature: Original Retrieve article from Numdam
XI: 23, 365-375, LNM 581 (1977)
DELLACHERIE, Claude;
STRICKER, Christophe
Changements de temps et intégrales stochastiques (
Martingale theory)
A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see
925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also
1135Comment: The last section states an interesting open problem
Keywords: Changes of time,
Spectral representationNature: Original Retrieve article from Numdam
XI: 24, 376-382, LNM 581 (1977)
DOLÉANS-DADE, Catherine;
MEYER, Paul-André
Équations différentielles stochastiques (
Stochastic calculus)
This is an improved and simplified exposition of the existence and uniqueness theorem for solutions of stochastic differential equations with respect to semimartingales, as proved by the first author in
Zeit. für W-theorie, 36, 1976 and by Protter in
Ann. Prob. 5, 1977. The theory has become now so classical that the paper has only historical interest
Keywords: Stochastic differential equations,
SemimartingalesNature: Exposition,
Original additions Retrieve article from Numdam
XI: 25, 383-389, LNM 581 (1977)
DOLÉANS-DADE, Catherine;
MEYER, Paul-André
Une caractérisation de $BMO$ (
Martingale theory)
Kazamaki gave in
1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention
Comment: Related subjects occur in
1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible
Keywords: $BMO$,
Stochastic exponentials,
Martingale inequalitiesNature: Original Retrieve article from Numdam
XI: 26, 390-410, LNM 581 (1977)
JACOD, Jean
Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales (
Martingale theory)
This paper develops the theory of stochastic integration (previsible and optional) with respect to local martingales starting from the particular case of continuous local martingales, and from the explicit description of the jumps of a local martingale (
1121,
1129). Then the theory of stable subspaces of $H^1$ (instead of the usual $H^2$) is developed, as well as the stochastic integral with respect to a random measure. A characterization is given of the jump process of a semimartingale. Then previsible stochastic integrals for semimartingales are given a maximal extension, and optional integrals for semimartingales (differing as usual from those for martingales) are defined
Comment: On the maximal extension of the stochastic integral $H{\cdot}X$ with $H$ previsible, see also Jacod,
Calcul stochastique et problèmes de martingales, Springer 1979. Other, equivalent, definitions are given in
1415,
1417,
1424 and
1530Keywords: Stochastic integrals,
Optional stochastic integrals,
Random measures,
SemimartingalesNature: Original Retrieve article from Numdam
XI: 27, 411-414, LNM 581 (1977)
KOSKAS, Maurice
Images d'équations différentielles stochastiques (
Stochastic calculus)
This paper answers a natural question: can one take computations performed on ``canonical'' versions of processes back to their original spaces? It is related to Stricker's work (
Zeit. für W-theorie, 39, 1977) on the restriction of filtrations
Keywords: Stochastic differential equationsNature: Original Retrieve article from Numdam
XI: 28, 415-417, LNM 581 (1977)
LENGLART, Érik
Une caractérisation des processus prévisibles (
General theory of processes)
One of the results of this short paper is the following: a bounded optional process $X$ is previsible if and only if, for every martingale $M$ of integrable variation, the Stieltjes integral process $X\sc M$ is a martingale
Keywords: Previsible processesNature: Original Retrieve article from Numdam
XI: 29, 418-434, LNM 581 (1977)
LÉPINGLE, Dominique
Sur la représentation des sauts des martingales (
Martingale theory)
The problem discussed in this paper consists in decomposing into two parts a local martingale, so that one part has its jumps contained in a given thin optional set $D$ and the other one is continuous on $D$. The main theorem of
1121 is proved independently as an important technical tool
Comment: See also
1335Keywords: Local martingales,
Jumps,
Optional stochastic integralsNature: Original Retrieve article from Numdam
XI: 30, 435-445, LNM 581 (1977)
MAISONNEUVE, Bernard
Une mise au point sur les martingales locales continues définies sur un intervalle stochastique (
Martingale theory)
The following definition is given of a continuous local martingale $M$ on an open interval $[0,T[$, for an arbitrary stopping time $T$: two sequences are assumed to exist, one of stopping times $T_n\uparrow T$, one $(M_n)$ of continuous martingales, such that $M=M_n$ on $[0,T_n[$. Stochastic integration is studied, and the change of variable formula is extended. It is proved that the set where the limit $M_{T-}$ exists and is finite is a.s. the same as that where $\langle M,M\rangle_T<\infty$, a result whose proof under the usual definition (i.e., assuming $T$ is previsible) was not clear
Keywords: Martingales on a random set,
Stochastic integralsNature: Original Retrieve article from Numdam
XI: 31, 446-481, LNM 581 (1977)
MEYER, Paul-André
Notes sur les intégrales stochastiques (
Martingale theory)
This paper contains six additions to
1017. Chapter~I concerns Hilbert space valued martingales, following Métivier, defining in particular their operator valued brackets and the corresponding stochastic integrals. Chapter~II gives a new proof (due to Yan, and now classical) of the basic result on the structure of local martingales. Chapter~III is a theorem of Herz (and Lépingle in continuous time) on the representation of $BMO$ which corresponds to the ``maximal'' definition of $H^1$. Chapter~IV states that, if $(B_t)$ is a $BMO$ martingale and $(X_t)$ is a martingale bounded in $L^p$, then $\sup_t X^{\ast}_t |B_{\infty}-B_t|$ is also in $L^p$ with a norm controlled by that of $X$ ($1< p<\infty$; there is at least a wrong statement about $p=1$ at the bottom of p. 470). This result can be interpreted as $L^p$ boundedness of the commutator of two operators: multiplication by an element of $BMO$, and stochastic integration by a bounded previsible process. Chapter~V (again on $BMO$) has a wrong proof, and seems to be still an open problem. Chapter~VI consists of small additions and corrections, and in particular acknowledges the priority of P.W.~Millar for useful results on local times
Comment: Three errors are corrected in
1248 and
1249Keywords: Stochastic integrals,
Hilbert space valued martingales,
Operator stochastic integrals,
$BMO$Nature: Original Retrieve article from Numdam
XI: 32, 482-489, LNM 581 (1977)
MEYER, Paul-André
Sur un théorème de C. Stricker (
Martingale theory)
Some emphasis is put on a technical lemma used by Stricker to prove the well-known result that semimartingales remain so under restriction of filtrations (provided they are still adapted). The result is that a semimartingale up to infinity can be sent into the Hardy space $H^1$ by a suitable choice of an equivalent measure. This leads also to a simple proof and an extension of Jacod's theorem that the set of semimartingale laws is convex
Comment: A gap in a proof is filled in
1251Keywords: Hardy spaces,
Changes of measureNature: Original Retrieve article from Numdam
XI: 33, 490-492, LNM 581 (1977)
WALSH, John B.
A property of conformal martingales (
Martingale theory)
Almost every path of a (complex) conformal martingale on the open time interval $]0,\infty[$ has the following behaviour at time $0$: either it has a limit in the Riemann sphere, or it is everywhere dense
Comment: See also
1408Keywords: Conformal martingalesNature: Original Retrieve article from Numdam
XI: 34, 493-501, LNM 581 (1977)
YOR, Marc
A propos d'un lemme de Ch. Yoeurp (
General theory of processes,
Martingale theory)
Yoeurp's lemma is the following: if $A$ is a previsible process of bounded variation, its square bracket $[A,L]$ with any local martingale $L$ is a local martingale. This useful result was not easily accessible, thus a complete proof is given, with several new applications---in particular, this characterizes previsible processes of bounded variation among semimartingales
Keywords: Yoeurp's lemma,
Square bracketNature: Original Retrieve article from Numdam
XI: 35, 502-517, LNM 581 (1977)
YOR, Marc
Remarques sur la représentation des martingales comme intégrales stochastiques (
Martingale theory)
The main result on the relation between the previsible representation property of a set of local martingales and the extremality of their joint law appeared in a celebrated paper of Jacod-Yor,
Z. für W-theorie, 38, 1977. Several concrete applications are given here, in particular a complete proof of a ``folklore'' result on the representation of local martingales of a Lévy process, and a discussion of the commutation problem of
1123Comment: This is an intermediate paper between the Jacod-Yor results and the definitive version of previsible representation, using the theorem of Douglas, for which see
1221Keywords: Previsible representation,
Extreme points,
Independent increments,
Lévy processesNature: Original Retrieve article from Numdam
XI: 36, 518-528, LNM 581 (1977)
YOR, Marc
Sur quelques approximations d'intégrales stochastiques (
Martingale theory)
The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form
Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe
Publ. RIMS, Kyoto Univ. 15, 1979; Bismut, Mécanique Aléatoire, Springer LNM~866, 1981; Meyer
1505Keywords: Stochastic integrals,
Riemann sums,
Stratonovich integralsNature: Original Retrieve article from Numdam
XI: 37, 529-538, LNM 581 (1977)
MAISONNEUVE, Bernard
Changement de temps d'un processus markovien additif (
Markov processes)
A Markov additive process $(X_t,S_t)$ (Cinlar,
Z. für W-theorie, 24, 1972) is a generalisation of a pair $(X,S)$ where $X$ is a Markov process with arbitrary state space, and $S$ is an additive functional of $X$: in the general situation $S$ is positive real valued, $X$ is a Markov process in itself, and the pair $(X,S)$ is a Markov processes, while $S$ is an additive functional
of the pair. For instance, subordinators are Markov additive processes with trivial $X$. A simpler proof of a basic formula of Cinlar is given, and it is shown also that a Markov additive process gives rise to a regenerative system in a slightly extended sense
Comment: See also
1513Keywords: Markov additive processes,
Additive functionals,
Regenerative sets,
Lévy systemsNature: Original Retrieve article from Numdam
XII: 01, 1-19, LNM 649 (1978)
PRATELLI, Maurizio
Une version probabiliste d'un théorème d'interpolation de G. Stampacchia (
Martingale theory,
Functional analysis)
This theorem is similar to the Marcinkievicz interpolation theorem, in the sense that at one endpoint a weak $L^p$ inequality is involved, but at the other endpoint the spaces involved are some $L^p$ and $BMO$. It concerns linear operators only, not sublinear ones like the Marcinkiewicz theorem. A closely related result, concerning the discrete-time case, had been proved earlier by Stroock,
Comm Pure Appl. Math.,
26, 1973
Keywords: Interpolation,
$BMO$Nature: Original Retrieve article from Numdam
XII: 02, 20-21, LNM 649 (1978)
STRICKER, Christophe
Une remarque sur les changements de temps et les martingales locales (
Martingale theory)
It is well known (see
606) that in general the class of local martingales is not invariant under changes of time. Here it is shown that, if ${\cal F}_0$ is trivial, a process which remains a local martingale under all changes of time (with bounded stopping times) is a true martingale (in full generality, it is so conditionally to ${\cal F}_0$)
Keywords: Changes of time,
Weak martingalesNature: Original Retrieve article from Numdam
XII: 03, 22-34, LNM 649 (1978)
JACOD, Jean
Projection prévisible et décomposition multiplicative d'une semi-martingale positive (
General theory of processes)
The problem discussed is the decomposition of a positive ($\ge0$) special semimartingale $X$ (the most interesting cases being super- and submartingales) into a product of a positive local martingale and a positive previsible process of finite variation. The problem is solved here in the greatest possible generality, on a maximal non-vanishing domain for $X$---this is a previsible stochastic interval $[0,S)$ which at $S$ may be open or closed
Comment: This papers improves on
1021 and
1023Keywords: Semimartingales,
Multiplicative decompositionNature: Original Retrieve article from Numdam
XII: 04, 35-46, LNM 649 (1978)
MÉMIN, Jean
Décompositions multiplicatives de semimartingales exponentielles et applications (
General theory of processes)
It is shown that, given two semimartingales $U,V$ such that $U$ has no jump equal to $-1$, there is a unique semimartingale $X$ such that ${\cal E}(X)\,{\cal E}(U)={\cal E}(V)$. This result is applied to recover all known results on multiplicative decompositions
Comment: The results of this paper are used in Mémin-Shiryaev
1312Keywords: Stochastic exponentials,
Semimartingales,
Multiplicative decompositionNature: Original Retrieve article from Numdam
XII: 05, 47-50, LNM 649 (1978)
KAZAMAKI, Norihiko
A remark on a problem of Girsanov (
Martingale theory)
It is shown that, if $M$ is a continuous local martingale which belongs to $BMO$, its stochastic exponential is a uniformly integrable martingale
Comment: This has become a well-known result. It is false for complex valued martingales, even bounded ones: see
1832Keywords: Stochastic exponentials,
$BMO$Nature: Original Retrieve article from Numdam
XII: 06, 51-52, LNM 649 (1978)
GARCIA, M.;
MAILLARD, P.;
PELTRAUT, Y.
Une martingale de saut multiplicatif donné (
Martingale theory)
Given a totally inaccessible stopping time $T$, it is shown how to construct a strictly positive martingale $M$ with $M_0=1$, such that its only jump occurs at time $T$ and $M_T/M_{T-}=K$, a strictly positive constant
Comment: See also
1308Keywords: Totally inaccessible stopping timesNature: Original Retrieve article from Numdam
XII: 07, 53-56, LNM 649 (1978)
LENGLART, Érik
Sur la localisation des intégrales stochastiques (
Stochastic calculus)
A mapping $T$ from processes to processes is
local if, whenever two processes $X,Y$ are equal on an event $A\subset\Omega$, the same is true for $TX,TY$. Classical results on locality in stochastic calculus are derived here in a simple way from the generalized Girsanov theorem (which concerns a pair of laws $P,Q$ with $Q$ absolutely continuous with respect to $P$, but not necessarily equivalent to it: see Lenglart,
Zeit. für W-theorie, 39, 1977). A new result is derived: if $X$ and $Y$ are semimartingales and their difference is of finite variation on an event $A$, then their continuous martingale parts are equal on $A$
Keywords: Girsanov's theoremNature: Original Retrieve article from Numdam
XII: 08, 57-60, LNM 649 (1978)
MEYER, Paul-André
Sur un théorème de J. Jacod (
General theory of processes)
Consider a given process $X$ adapted to a given filtration $({\cal F}_t)$. The set of laws of semimartingales consists of those laws $P$ under which $X$ is a semimartingale with respect to $({\cal F}_t)$ suitably completed. Jacod proved that the set of laws of semimartingales is convex. This is extended here to countable convex combinations, and to integrals
Comment: This easy paper has some historical interest, as it raised the problem of initial enlargement of a filtration
Keywords: Semimartingales,
Enlargement of filtrations,
Laws of semimartingalesNature: Original Retrieve article from Numdam
XII: 09, 61-69, LNM 649 (1978)
YOR, Marc
Grossissement d'une filtration et semi-martingales~: théorèmes généraux (
General theory of processes)
Given a filtration $({\cal F}_t)$ and a positive random variable $L$, the so-called
progressively enlarged filtration is the smallest one $({\cal G}_t)$ containing $({\cal F}_t)$, and for which $L$ is a stopping time. The enlargement problem consists in describing the semimartingales $X$ of ${\cal F}$ which remain semimartingales in ${\cal G}$, and in computing their semimartingale characteristics. In this paper, it is proved that $X_tI_{\{t< L\}}$ is a semimartingale in full generality, and that $X_tI_{\{t\ge L\}}$ is a semimartingale whenever $L$ is
honest for $\cal F$, i.e., is the end of an $\cal F$-optional set
Comment: This result was independently discovered by Barlow,
Zeit. für W-theorie, 44, 1978, which also has a huge intersection with
1211. Complements are given in
1210, and an explicit decomposition formula for semimartingales in
1211Keywords: Enlargement of filtrations,
Honest timesNature: Original Retrieve article from Numdam
XII: 10, 70-77, LNM 649 (1978)
DELLACHERIE, Claude;
MEYER, Paul-André
A propos du travail de Yor sur le grossissement des tribus (
General theory of processes)
This paper adds a few comments and complements to the preceding one
1209; for instance, the enlargement map is bounded in $H^1$
Keywords: Enlargement of filtrations,
Honest timesNature: Original Retrieve article from Numdam
XII: 11, 78-97, LNM 649 (1978)
JEULIN, Thierry;
YOR, Marc
Grossissement d'une filtration et semi-martingales~: Formules explicites (
General theory of processes)
This contains very substantial improvements on
1209, namely, the explicit computation of the characteristics of the semimartingales involved
Comment: For additional results on enlargements, see the two Lecture Notes volumes
833 (T. Jeulin) and
1118. See also
1350Keywords: Enlargement of filtrations,
Honest timesNature: Original Retrieve article from Numdam
XII: 12, 98-113, LNM 649 (1978)
DELLACHERIE, Claude;
MEYER, Paul-André;
YOR, Marc
Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (
Martingale theory,
Functional analysis)
The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see
1519; see also
3021 and
3316)
Keywords: Hardy spaces,
$BMO$Nature: Original Retrieve article from Numdam
XII: 13, 114-131, LNM 649 (1978)
YAMADA, Toshio
Sur une construction des solutions d'équations différentielles stochastiques dans le cas non-lipschitzien (
Stochastic calculus)
The results of this paper improve on those of the author's paper (
Zeit. für W-theorie, 36, 1976) concerning a one-dimensional stochastic differential equations of the classical Ito type, whose coefficients satisfy a Hölder-like condition instead of the standard Lipschitz condition. The proofs are simplified, and strong convergence of the Cauchy method is shown
Comment: Such equations play an important role in the theory of Bessel processes (see chapter XI of Revuz-Yor,
Continuous Martingales and Brownian Motion, Springer 1999
Keywords: Stochastic differential equations,
Hölder conditionsNature: Original Retrieve article from Numdam
XII: 14, 132-133, LNM 649 (1978)
CHOU, Ching Sung
Extension au cas continu d'un théorème de Dubins (
Martingale theory)
After a suitable translation, Dubins' theorem can be stated as follows: if $X$ is a positive submartingale with $X_t\in L^2$ for all $t$, then $X^2-[X,X]$ is a submartingale
Keywords: SubmartingalesNature: Original Retrieve article from Numdam
XII: 15, 134-137, LNM 649 (1978)
LÉPINGLE, Dominique
Une inégalité de martingales (
Martingale theory)
The following inequality for a discrete time adapted process $(a_n)$ and its conditional expectations $b_n=E[a_n\,|\,{\cal F}_{n-1}]$ is proved: $$\|(\sum_n b_n^2)^{1/2}\|_1\le 2\|(\sum_n a_n^2)^{1/2}\|_1\ .$$ A similar inequality in $L^p$, $1\!<\!p\!<\!\infty$, does not require adaptedness, and is due to Stein
Keywords: Inequalities,
Quadratic variationNature: Original Retrieve article from Numdam
XII: 16, 138-147, LNM 649 (1978)
LÉPINGLE, Dominique
Sur certains commutateurs de la théorie des martingales (
Martingale theory)
Let $\beta$ the operator on (closed) martingales $X$ consisting in multiplication of $X_{\infty}$ by a given r.v. $B$. One investigates the commutator $J\beta-\beta J$ of $\beta$ with some operator $J$ on martingales (a typical example is stochastic integration $JX=H.X$ where $H$ is a given bounded previsible process), expecting this commutator to be bounded in $L^p$ if $B$ belongs to $BMO$. This is indeed true under natural conditions on $J$
Keywords: $BMO$Nature: Original Retrieve article from Numdam
XII: 17, 148-161, LNM 649 (1978)
LÉPINGLE, Dominique
Sur le comportement asymptotique des martingales locales (
Martingale theory)
This paper is devoted to the extension of well-known statements (the strong law of large numbers, the Borel-Cantelli lemma, and the easier half of the law of the iterated logarithm) to right-continuous local martingales. An interesting technical point is the definition of a family of exponential supermartingales
Keywords: Law of large numbers,
Borel-Cantelli lemma,
Exponential martingales,
Law of the iterated logarithmNature: Original Retrieve article from Numdam
XII: 18, 162-169, LNM 649 (1978)
CAIROLI, Renzo
Une représentation intégrale pour les martingales fortes (
Several parameter processes)
This paper uses the results of Cairoli-Walsh,
Ann. Prob. 5, 1977, to prove a stochastic integral representation of the strong martingales of the Brownian sheet filtration, without assuming they are square integrable
Keywords: Strong martingales,
Brownian sheetNature: Original Retrieve article from Numdam
XII: 19, 170-179, LNM 649 (1978)
MÉTRAUX, C.
Quelques inégalités pour martingales à paramètre bidimensionnel (
Several parameter processes)
This paper extends to two-parameter discrete martingales the classical Burkholder inequalities ($1<p<\infty$) and a few more inequalities
Keywords: Burkholder inequalitiesNature: Original Retrieve article from Numdam
XII: 20, 180-264, LNM 649 (1978)
BISMUT, Jean-Michel
Contrôle des systèmes linéaires quadratiques~: applications de l'intégrale stochastique (
Control theory)
(To be completed) This is a set of lectures in control theory, which makes use of refined techniques in stochastic integration. It should be reviewed in detail
Comment: To be completed
Keywords: Optional stochastic integralsNature: Original Retrieve article from Numdam
XII: 21, 265-309, LNM 649 (1978)
YOR, Marc;
SAM LAZARO, José de
Sous-espaces denses dans $L^1$ ou $H^1$ et représentation des martingales (
Martingale theory)
This paper was a considerable step in the study of the general martingale problem, i.e., of the set ${\cal P}$ of all laws on a filtered measurable space under which a given set ${\cal N}$ of (adapted, right continuous) processes are local martingales. The starting point is a theorem from measure theory due to R.G. Douglas (
Michigan Math. J. 11, 1964), and the main technical difference with preceding papers is the systematic use of stochastic integration in $H^1$. The main result can be stated as follows: given a law $P\in{\cal P}$, the set ${\cal N}$ has the previsible representation property, i.e., ${\cal F}_0$ is trivial and stochastic integrals with respect to elements of ${\cal N}$ are dense in $H^1$, if and only if $P$ is an extreme point of ${\cal P}$. Many examples and applications are given
Comment: The second named author's contribution concerns only the appendix on homogeneous martingales
Keywords: Previsible representation,
Douglas theorem,
Extremal lawsNature: Original Retrieve article from Numdam
XII: 22, 310-331, LNM 649 (1978)
WILLIAMS, David
The Q-matrix problem 3: The Lévy-kernel problem for chains (
Markov processes)
After solving the Q-matrix problem in
1014, the author constructs here a Markov chain from a Q-matrix on a countable space $I$ which satisfies several desirable conditions. Among them, the following: though the process is defined on a (Ray) compactification of $I$, the Q-matrix should describe the full Lévy kernel. Otherwise stated, whenever the process jumps, it does so from a point of $I$ to a point of $I$. The construction is extremely delicate
Keywords: Markov chainsNature: Original Retrieve article from Numdam
XII: 23, 332-341, LNM 649 (1978)
BRETAGNOLLE, Jean;
HUBER, Catherine
Lois empiriques et distance de Prokhorov (
Mathematical statistics)
Let $F$ be a distribution function, and $F_n$ be the corresponding (random) empirical distribution functions. Let $d$ be a distance on the set of distribution functions. The problem is the speed of convergence of $F_n$ to $F$, i.e., to find the exponent $\alpha$ such that $P(n^{\alpha}d(F_n,F)>u)$ remains bounded and bounded away from $0$ for some $u>0$. The distance used is that of Prohorov, for which auxiliary results are proved. It is shown that the exponent lies between 1/3 and 1/2, the latter case being that of regular distribution functions, but the whole interval being possible for sufficiently singular ones
Keywords: Empirical distribution function,
Prohorov distanceNature: Original Retrieve article from Numdam
XII: 24, 342-363, LNM 649 (1978)
BRETAGNOLLE, Jean;
HUBER, Catherine
Estimation des densités~: risque minimax (
Mathematical statistics)
A sequel to the preceding paper
1223. The speed of convergence in the estimation of the density of a law $f$ from the observation of a sample is discussed
Comment: For a correction see
1360. An improved version appeared in (
Zeit. für W-theorie, 47, 1979)
Keywords: Empirical distribution functionNature: Original Retrieve article from Numdam
XII: 25, 364-377, LNM 649 (1978)
STRICKER, Christophe
Les ralentissements en théorie générale des processus (
General theory of processes)
Given a filtration $({\cal F}t)$ and a stopping time $T$, we may define a new filtration $({\cal G}_t)$ as follows: we introduce an independent random variable $S$, and in intuitive language, we run the picture of $({\cal F}_t)$ up to time $T$, freeze the image between times $T$ and $T+S$, and then start running it again. The main result of this paper is the possibility, by performing this at all the times of discontinuity of $({\cal F}_t)$, to construct a filtration $({\cal G}_t)$ which is quasi-left-continuous. Though the idea is simple, there are considerable technical difficulties
Nature: Original Retrieve article from Numdam
XII: 26, 378-397, LNM 649 (1978)
BROSSARD, Jean
Comportement non-tangentiel et comportement brownien des fonctions harmoniques dans un demi-espace. Démonstration probabiliste d'un théorème de Calderon et Stein (
Potential theory,
Real analysis)
Given a harmonic function $u$ in a half space, Stein (
Acta Math. 106, 1961) shows that the boundary points $x$ such that 1) $u$ has a non-tangential limit at $x$, 2) $u$ is ``non tangentially bounded'' near $x$, 3) $\nabla u$ is locally $L^2$ in the non-tangential cones at $x$, are the sames, except for sets of measure $0$. This result is given here a probabilistic proof using conditional Brownian motion
Keywords: Harmonic functions in a half-space,
Non-tangential limitsNature: Original Retrieve article from Numdam
XII: 27, 398-410, LNM 649 (1978)
GETOOR, Ronald K.
Homogeneous potentials (
General theory of processes)
This is a development in Knight's prediction theory as described in
1007,
1008. Let $(Z_t^\mu)$ be the prediction process associated with a given measure $\mu$. Then it is shown that a bounded homogeneous right continuous supermartingale (or potential) under $\mu$ remains so under the measures $Z_t^\mu$
Keywords: Prediction theoryNature: Original Retrieve article from Numdam
XII: 28, 411-423, LNM 649 (1978)
MEYER, Paul-André
Convergence faible et compacité des temps d'arrêt, d'après Baxter et Chacón (
General theory of processes)
Baxter and Chacón (
Zeit. für W-theorie, 40, 1977) introduced a topology on the sets of ``fuzzy'' times and of fuzzy stopping times which turn these sets into compact metrizable spaces---a fuzzy r.v. $T$ is a right continuous decreasing process $M_t$ with $M_{0-}=1$, $M_t(\omega)$ being interpreted for each $\omega$ as the distribution function $P_{\omega}\{T>t\}$. When this process is adapted the fuzzy r.v. is a fuzzy stopping time. A number of properties of this topology are investigated
Comment: See
1536 for an extension to Polish spaces
Keywords: Stopping times,
Fuzzy stopping timesNature: Exposition,
Original additions Retrieve article from Numdam
XII: 29, 424-424, LNM 649 (1978)
DELLACHERIE, Claude
Convergence en probabilité et topologie de Baxter-Chacón (
General theory of processes)
It is shown that on the set of ordinary stopping times, the Baxter-Chacón topology is simply convergence in probability
Keywords: Stopping timesNature: Original Retrieve article from Numdam
XII: 30, 425-427, LNM 649 (1978)
DELLACHERIE, Claude;
MEYER, Paul-André
Construction d'un processus prévisible ayant une valeur donnée en un temps d'arrêt (
General theory of processes)
Let $T$ be a stopping time, $X$ be an integrable r.v., and put $A_t=I_{\{t\ge T\}}$ and $B_t=XA_t$. Then the previsible compensator $(\tilde B_t)$ has a previsible density $Z_t$ with respect to $(\tilde A_t)$, whose value $Z_T$ at time $T$ is $E[X\,|\,{\cal F}_{T-}]$, and in particular if $X$ is ${\cal F}_T$-measurable it is equal to $X$
Keywords: Stopping timesNature: Original Retrieve article from Numdam
XII: 31, 428-445, LNM 649 (1978)
KNIGHT, Frank B.
On the sojourn times of killed Brownian motion (
Brownian motion)
To be completed
Keywords: Sojourn times,
Laplace transformsNature: Original Retrieve article from Numdam
XII: 32, 446-456, LNM 649 (1978)
TAYLOR, John C.
Some remarks on Malliavin's comparison lemma and related topics (
Diffusion theory)
The comparison lemma considered here gives estimates for the hitting probabilities of a several dimensional diffusion in terms of the hitting probabilities of a half line for suitably constructed one-dimensional diffusions. A self-contained proof is given
Keywords: Hitting probabilitiesNature: Original Retrieve article from Numdam
XII: 33, 457-467, LNM 649 (1978)
MAINGUENEAU, Marie Anne
Temps d'arrêt optimaux et théorie générale (
General theory of processes)
This is a general discussion of optimal stopping in continuous time. Fairly advanced tools like strong supermartingales, Mertens' decomposition are used
Comment: The subject is taken up in
1332Keywords: Optimal stopping,
Snell's envelopeNature: Original Retrieve article from Numdam
XII: 35, 482-488, LNM 649 (1978)
YOR, Marc;
MEYER, Paul-André
Sur l'extension d'un théorème de Doob à un noyau $\sigma$-fini, d'après G. Mokobodzki (
Measure theory)
Given a kernel $K(x,dy)$ consisting of probability measures, all of them absolutely continuous with respect to a measure $\mu$, Doob proved long ago using martingale theory that $K(x,dy)=k(x,y)\,\mu(dy)$ with a jointly measurable density $k(x,y)$. What happens if the measures $K(x,dy)$ are $\sigma$-finite? The answer is that Doob's result remains valid if $K$, considered as a mapping $x\mapsto K(x,\,.\,)$ taking values in the set of all $\sigma$-finite measures absolutely continuous w.r.t. $\mu$ (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability
Comment: The subject is discussed further in
1527. Note a mistake near the bottom of page 486: the $\sigma$-field on $E$ should be associated with the
weak topology of $L[\infty$, or with the topology of $L^0$
Keywords: Kernels,
Radon-Nikodym theoremNature: Original Retrieve article from Numdam
XII: 36, 489-490, LNM 649 (1978)
MOKOBODZKI, Gabriel
Domination d'une mesure par une capacité (
Measure theory)
A bounded measure $\mu$ is said to be dominated by a capacity $C$ (countably subadditive, continuous along increasing sequences; neither strong subadditivity nor decreasing sequences are mentioned) if all sets of capacity $0$ have also measure $0$. The main result then states that the space can be decomposed into a set $A_0$ of capacity $0$, and disjoint sets $A_n$ on each of which $\mu$ is smaller than a multiple of $C$
Keywords: Radon-Nikodym theorem,
CapacitiesNature: Original Retrieve article from Numdam
XII: 37, 491-508, LNM 649 (1978)
MOKOBODZKI, Gabriel
Ensembles à coupes dénombrables et capacités dominées par une mesure (
Measure theory,
General theory of processes)
Let $X$ be a compact metric space $\mu$ be a bounded measure. Let $F$ be a given Borel set in $X\times
R_+$. For $A\subset X$ define $C(A)$ as the outer measure of the projection on $X$ of $F\cap(A\times
R_+)$. Then it is proved that, if there is some measure $\lambda$ such that $\lambda$-null sets are $C$-null (the relation goes the reverse way from the preceding paper
1236!) then $F$ has ($\mu$-a.s.) countable sections, and if the property is strengthened to an $\epsilon-\delta$ ``absolute continuity'' relation, then $F$ has ($\mu$-a.s.) finite sections
Comment: This was a long-standing conjecture of Dellacherie (
707), suggested by the theory of semi-polar sets. For further development see
1602Keywords: Sets with countable sectionsNature: Original Retrieve article from Numdam
XII: 38, 509-511, LNM 649 (1978)
DELLACHERIE, Claude
Appendice à l'exposé de Mokobodzki (
Measure theory,
General theory of processes)
Some comments on
1237: a historical remark, a relation with a result of Talagrand, the inclusion of a converse (due to Horowitz) to the case of finite sections, and the solution to the conjecture from
707Keywords: Sets with countable sections,
Semi-polar setsNature: Original Retrieve article from Numdam
XII: 39, 512-514, LNM 649 (1978)
DELLACHERIE, Claude
Sur l'existence de certains ess.inf et ess.sup de familles de processus mesurables (
General theory of processes)
The word ``essential'' in the title refers to inequalities between processes up to evanescent sets. Since in the case of a probability space consisting of one point, this means inequalities everywhere, it is clear that additional assumptions are necessary. Such essential bounds are shown to exist whenever the sample functions are upper semicontinuous in the right topology, or the left topology (and of course also if they are lower semicontinuous). This covers in particular the case of strong supermartingales and Snell's envelopes
Keywords: Essential suprema,
Evanescent setsNature: Original Retrieve article from Numdam
XII: 40, 515-522, LNM 649 (1978)
DELLACHERIE, Claude
Supports optionnels et prévisibles d'une P-mesure et applications (
General theory of processes)
A $P$-measure is a measure on $\Omega\times
R_+$ which does not charge $P$-evanescent sets. A $P$-measure has optional and previsible projections which are themselves $P$-measures. As usual, supports are minimal sets carrying a measure, possessing different properties like being optional/previsible, being right/left closed. The purpose of the paper is to find out which kind of supports do exist. Applications are given to honest times
Comment: See
1339 for a complement concerning honest times
Keywords: Projection theorems,
Support,
Honest timesNature: Original Retrieve article from Numdam
XII: 42, 524-563, LNM 649 (1978)
HILLARD, Gérard
Exemples de normes en théorie descriptive des ensembles (
Descriptive set theory)
The situations described in this paper are special cases of
1104, where a coanalytic set $A$ was represented as the union of an increasing family $A_{\alpha}$ of analytic sets indexed by the countable ordinals, such that every analytic subset of $A$ is contained in some $A_{\alpha}$. The hypotheses of
1104 are not easy to check: they are shown here to include the classical Cantor derivation on the coanalytic space of countable compact sets, and a new example on the coanalytic space of all right continuous functions
Comment: The whole subject has been exposed anew in Chapter XXIV of Dellacherie-Meyer,
Probabilités et PotentielKeywords: Derivations (set-theoretic),
Kunen-Martin theoremNature: Original Retrieve article from Numdam
XII: 43, 564-566, LNM 649 (1978)
DELLACHERIE, Claude;
MOKOBODZKI, Gabriel
Deux propriétés des ensembles minces (abstraits) (
Descriptive set theory)
Given a class ${\cal S}$ of Borel sets understood as ``small'' sets, the class ${\cal L}$ consisting of their conplements understood as ``large'' sets, a set $A$ is said to be ${\cal S}$-thin if does not contain uncountably many disjoint ``large'' sets. For instance, if ${\cal S}$ is the class of polar sets, then thin sets are the same as semi-polar sets. Two general theorems are proved here on thin sets
Keywords: Thin sets,
Semi-polar sets,
Essential supremaNature: Original Retrieve article from Numdam
XII: 44, 567-690, LNM 649 (1978)
NANOPOULOS, Constantin;
NOBELIS, Photis
Régularité et propriétés limites des fonctions aléatoires (
Miscellanea,
Gaussian processes)
This paper extends to the non-Gaussian case methods to study the regularity of sample paths which have proved useful in the Gaussian case, notably that of majorizing measures (to be completed)
Comment: To be completed
Keywords: Sample path regularity,
Majorizing measuresNature: Original Retrieve article from Numdam
XII: 45, 691-706, LNM 649 (1978)
FERNIQUE, Xavier
Caractérisation de processus à trajectoires majorées ou continues (
Miscellanea,
Gaussian processes)
The methods which lead the author to necessary and sufficient conditions for boundedness or continuity of stationary Gaussian processes are extended and applied to non-stationary Gaussian processes and non-Gaussian processes
Keywords: Sample path regularityNature: Original Retrieve article from Numdam
XII: 46, 707-738, LNM 649 (1978)
DELLACHERIE, Claude
Théorie unifiée des capacités et des ensembles analytiques (
Descriptive set theory)
A Choquet capacity takes one set as argument and produces a number. Along the years, one has considered multicapacities (which take as arguments finitely many sets) and capacitary operators (which produce sets instead of numbers). The essential result of this paper is that, if one allows functions of infinitely many arguments which produce sets, then the corresponding ``Choquet theorem'' gives all the classical results at a time, without need of an independent theory of analytic sets
Comment: For a more systematic exposition, see Chapter XI of Dellacherie-Meyer
Probabilités et PotentielKeywords: Capacities,
Analytic setsNature: Original Retrieve article from Numdam
XII: 54, 742-745, LNM 649 (1978)
DELLACHERIE, Claude
Quelques applications du lemme de Borel-Cantelli à la théorie des semimartingales (
Martingale theory,
Stochastic calculus)
The general idea is the following: many constructions relative to one single semimartingale---like finding a sequence of stopping times increasing to infinity which reduce a local martingale, finding a change of law which sends a given semimartingale into $H^1$ or $H^2$ (locally)---can be strengthened to handle at the same time countably many given semimartingales
Nature: Original Retrieve article from Numdam
XII: 55, 746-756, LNM 649 (1978)
DELLACHERIE, Claude
Quelques exemples familiers, en probabilités, d'ensembles analytiques non boréliens (
Descriptive set theory,
General theory of processes)
There is a tendency to consider that the naive, healthy probabilist should keep away from unnecessary abstraction, and in particular from analytic sets which are not Borel. This paper shows that such sets crop into probability theory in the most natural way. For instance, while the sample space of right-continuous paths with left limits is Borel, that of right-continuous paths without restriction on the left is coanalytic and non-Borel. Also, on the Borel sample space of right-continuous paths with left limits, the hitting time of a closed set is a function which is coanalytic and non-Borel
Keywords: Analytic setsNature: Original Retrieve article from Numdam
XII: 56, 757-762, LNM 649 (1978)
MEYER, Paul-André
Inégalités de normes pour les intégrales stochastiques (
Stochastic calculus)
Inequalities of the following kind were introduced by Émery: $$\|X.M\|_{H^p}\le c_p \| X\|_{S^p}\,\| M\|$$ where the left hand side is a stochastic integral of the previsible process $X$ w.r.t. the semimartingale $M$, $S^p$ is a supremum norm, and the norm $H^p$ for semimartingales takes into account the Hardy space norm for the martingale part and the $L^p$ norm of the total variation for the finite variation part. On the right hand side, Émery had used a norm called $H^{\infty}$. Here a weaker $BMO$-like norm for semimartingales is suggested
Keywords: Stochastic integrals,
Hardy spacesNature: Original Retrieve article from Numdam
XII: 58, 770-774, LNM 649 (1978)
MEYER, Paul-André
Sur le lemme de La Vallée Poussin et un théorème de Bismut (
Measure theory,
General theory of processes)
Bismut proved that every optional process which belongs to the class (D) is the optional projection of a (non-adapted) process whose supremum is in $L^1$. This is given a more precise form, using the relation between uniform integrability and moderate Orlicz spaces
Keywords: Uniform integrability,
Class (D) processes,
Moderate convex functionsNature: Exposition,
Original additions Retrieve article from Numdam
XII: 59, 775-803, LNM 649 (1978)
MEYER, Paul-André
Martingales locales fonctionnelles additives (two talks) (
Markov processes)
The purpose of the paper is to specialize the standard theory of Hardy spaces of martingales to the subspaces of additive martingales of a Markov process. The theory is not complete: the dual of (additive) $H^1$ seems to be different from (additive) $BMO$
Keywords: Hardy spaces,
Additive functionalsNature: Original Retrieve article from Numdam
XII: 60, 804-805, LNM 649 (1978)
LETTA, Giorgio
Un système de notations pour les processus de Markov (
Markov processes)
Instead of notations like $X_T$, $\Theta_T$, etc, the author suggests to use kernel notations---for instance, $X^T$ is the submarkovian kernel $f\mapsto f\circ X_T$ ($0$ on $\{T=\infty\}$). Then the main properties of Markov processes are expressed by simple kernel equalities
Nature: Original Retrieve article from Numdam
XIII: 01, 1-3, LNM 721 (1979)
BORELL, Christer
On the integrability of Banach space valued Walsh polynomials (
Banach space valued random variables)
The $L^2$ space over the standard Bernoulli measure on $\{-1,1\}^
N$ has a well-known orthogonal basis $(e_\alpha)$ indexed by the finite subsets of $
N$. The Walsh polynomials of order $d$ with values in a Banach space $E$ are linear combinations $\sum_\alpha c_\alpha e_\alpha$ where $c_\alpha\in E$ and $\alpha$ is a finite subset with $d$ elements. It is shown that on this space (as on the Wiener chaos spaces) all $L^p$ norms are equivalent with precise bounds, for $1<p<\infty$. The proof uses the discrete version of hypercontractivity
Keywords: Walsh polynomials,
HypercontractivityNature: Original Retrieve article from Numdam
XIII: 02, 4-21, LNM 721 (1979)
CHATTERJI, Shrishti Dhav
Le principe des sous-suites dans les espaces de Banach (
Banach space valued random variables)
The ``principle of subsequences'' investigated in the author's paper
604 says roughly that any suitably bounded sequence of r.v.'s contains a subsequence which in some respect ``looks like'' a sequence of i.i.d. random variables. Extensions are considered here in the case of Banach space valued random variables. The paper has the character of a preliminary investigation, though several non-trivial results are indicated (one of them in the Hilbert space case)
Keywords: SubsequencesNature: Original Retrieve article from Numdam
XIII: 03, 22-40, LNM 721 (1979)
GINÉ, Evarist
Domains of attraction in Banach spaces (
Banach space valued random variables)
To be completed
Comment: A correction is given as
1402Nature: Original Retrieve article from Numdam
XIII: 04, 41-71, LNM 721 (1979)
LAPRESTÉ, Jean-Thierry
Charges, poids et mesures de Lévy dans les espaces vectoriels localement convexes (
Banach space valued random variables)
To be completed
Nature: Original Retrieve article from Numdam
XIII: 05, 72-89, LNM 721 (1979)
MARCUS, Michael B.;
PISIER, Gilles
Random Fourier series on locally compact abelian groups (
Banach space valued random variables,
Harmonic analysis)
To be completed
Nature: Original Retrieve article from Numdam
XIII: 06, 90-115, LNM 721 (1979)
AZÉMA, Jacques;
YOR, Marc
Une solution simple au problème de Skorokhod (
Brownian motion)
An explicit solution is given to Skorohod's problem: given a distribution $\mu$ with mean $0$ and finite second moment $\sigma^2$, find a (non randomized) stopping time $T$ of a Brownian motion $(X_t)$ such that $X_T$ has the distribution $\mu$ and $E[T]=\sigma^2$. It is shown that if $S_t$ is the one-sided supremum of $X$ at time $t$, $T=\inf\{t:S_t\ge\psi(X_t)\}$ solves the problem, where $\psi(x)$ is the barycenter of $\mu$ restricted to $[x,\infty[$. The paper has several interesting side results, like explicit families of Brownian martingales, and a proof of the Ray-Knight theorem on local times
Comment: The subject is further investigated in
1356 and
1441. See also
1515. A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XIII: 07, 116-117, LNM 721 (1979)
ÉMERY, Michel;
STRICKER, Christophe
Démonstration élémentaire d'un résultat d'Azéma et Jeulin (
Martingale theory)
A short proof is given for the following result: given a positive supermartingale $X$ and $h>0$, the supermartingale $E[X_{t+h}\,|\,{\cal F}_t]$ belongs to the class (D). The original proof (
Ann. Inst. Henri Poincaré, 12, 1976) used Föllmer's measures
Keywords: Class (D) processesNature: Original Retrieve article from Numdam
XIII: 08, 118-125, LNM 721 (1979)
YOEURP, Chantha
Sauts additifs et sauts multiplicatifs des semi-martingales (
Martingale theory,
General theory of processes)
First of all, the jump processes of special semimartingales are characterized (using a result of
1121,
1129 on the jump processes of local martingales). Then a similar problem is solved for multiplicative jumps, a result which includes that of Garcia and al.
1206. A technical lemma characterizes optional processes, bounded in $L^1$, whose previsible projection vanishes
Keywords: Jump processesNature: Original Retrieve article from Numdam
XIII: 09, 126-131, LNM 721 (1979)
PRATELLI, Maurizio
Le support exact du temps local d'une martingale continue (
Martingale theory)
It is well known in the Brownian case that the zero set and the support of the local time are the same. For a continuous local martingale $(X_t)$ with zero set $H$ and local time $(L_t)$, it is shown that the support of $dL$ is exactly the perfect kernel of the boundary of $H$
Keywords: Local timesNature: Original Retrieve article from Numdam
XIII: 10, 132-137, LNM 721 (1979)
SIDIBÉ, Ramatoulaye
Martingales locales à accroissements indépendants (
Martingale theory,
Independent increments)
It is shown here that a process with (stationary) independent increments which is a local martingale must be a true martingale
Comment: The case of non-stationary increments is considered in
1544. See also the errata sheet of vol. XV
Keywords: Local martingales,
Lévy processesNature: Original Retrieve article from Numdam
XIII: 11, 138-141, LNM 721 (1979)
REBOLLEDO, Rolando
Décomposition des martingales locales et raréfaction des sauts (
Martingale theory)
The general topic underlying this paper is that of convergence in law of a sequence of local martingales $M^n$ to a continuous Gaussian local martingale, i.e., a result analogue to the Central Limit Theorem in the Skorohod topology. This rests on three properties: tightness, convergence of the processes $<M^n,M^n>_t$ to a deterministic process, and a property of ``rarefaction of jumps''. The paper is devoted to a general discussion of the latter property
Comment: A correction is given as
1430Keywords: Convergence in law,
TightnessNature: Original Retrieve article from Numdam
XIII: 12, 142-161, LNM 721 (1979)
MÉMIN, Jean;
SHIRYAEV, Albert N.
Un critère prévisible pour l'uniforme intégrabilité des semimartingales exponentielles (
Martingale theory)
A condition is given so that the stochastic exponential of a special semimartingale $X$ is a uniformly integrable process. It involves only the local characteristics of $X$, i.e., its previsible compensator, Lévy measure, and quadratic variation of the continuous martingale part. The proof rests on multiplicative decompositions, and known results in the case of martingales
Keywords: Stochastic exponentials,
Semimartingales,
Multiplicative decomposition,
Local characteristicsNature: Original Retrieve article from Numdam
XIII: 13, 162-173, LNM 721 (1979)
CAIROLI, Renzo
Sur la convergence des martingales indexées par ${\bf N}\times{\bf N}$ (
Several parameter processes)
For two parameter (discrete) martingales, it is known that uniform integrability does not imply a.s. convergence. But if all (discrete) martingale transforms by indicators of previsible sets are uniformly integrable, then a.s. convergence obtains
Keywords: Almost sure convergence,
Martingale transformsNature: Original Retrieve article from Numdam
XIII: 14, 174-198, LNM 721 (1979)
CAIROLI, Renzo;
GABRIEL, Jean-Pierre
Arrêt de certaines suites multiples de variables aléatoires indépendantes (
Several parameter processes,
Independence)
Let $(X_n)$ be independent, identically distributed random variables. It is known that $X_T/T\in L^1$ for all stopping times $T$ (or the same with $S_n=X_1+...+X_n$ replacing $X_n$) if and only if $X\in L\log L$. The problem is to extend this to several dimensions, $
N^d$ ($d>1$) replacing $
N$. Then a stopping time $T$ becomes a stopping point, of which two definitions can be given (the past at time $n$ being defined either as the past rectangle, or the complement of the future rectangle), and $|T|$ being defined as the product of the coordinates). The appropriate space then is $L\log L$ or $L\log^d L$ depending on the kind of stopping times involved. Also the integrability of the supremum of the processes along random increasing paths is considered
Keywords: Stopping points,
Random increasing pathsNature: Original Retrieve article from Numdam
XIII: 15, 199-203, LNM 721 (1979)
MEYER, Paul-André
Une remarque sur le calcul stochastique dépendant d'un paramètre (
General theory of processes)
Call a ``process'' a measurable function $X(u,t,\omega)$ where $t$ and $\omega$ are as usual and $u$ is a parameter ranging over some nice measurable space ${\cal U}$. Say that $X$ is evanescent if $X(.\,,\,.\,,\omega)\equiv0$ for a.a. $\omega$. The problem is to define previsible processes, and previsible projections defined up to evanescent sets. This is achieved following Jacod,
Zeit. für W-Theorie, 31, 1975. The main feature is the corresponding use of random measures, previsible random measures, and previsible dual projections
Keywords: Processes depending on a parameter,
Previsible processes,
Previsible projections,
Random measuresNature: Exposition,
Original additions Retrieve article from Numdam
XIII: 16, 204-215, LNM 721 (1979)
DOLÉANS-DADE, Catherine;
MEYER, Paul-André
Un petit théorème de projection pour processus à deux indices (
Several parameter processes)
This paper proves a previsible projection theorem in the case of two-parameter processes, with a two-parameter filtration satisfying the standard commutation property of Cairoli-Walsh. The idea is to apply successively the projection operation of
1315 to the two coordinates
Keywords: Previsible processes (several parameters),
Previsible projections,
Random measuresNature: Original Retrieve article from Numdam
XIII: 17, 216-226, LNM 721 (1979)
LETTA, Giorgio
Quasimartingales et formes linéaires associées (
General theory of processes,
Martingale theory)
This paper gives a definition of a quasimartingale $X$ different from the usual one using the stochastic variation: the linear functional $E[\int_0^{\infty} H_s dX_s]$ should be relatively bounded on the vector lattice (Riesz space) of elementary previsible processes. Many classical theorems have simple proofs or elegant interpretations in this language
Keywords: Quasimartingales,
Riesz spacesNature: Original Retrieve article from Numdam
XIII: 18, 227-232, LNM 721 (1979)
BRUNEAU, Michel
Sur la $p$-variation d'une surmartingale continue (
Martingale theory)
The $p$-variation of a deterministic function being defined in the obvious way as a supremum over all partitions, the sample functions of a continuous martingale (and therefore semimartingale) are known to be of finite $p$-variation for $p>2$ (not for $p=2$ in general: non-anticipating partitions are not sufficient to compute the $p$-variation). If $X$ is a continuous supermartingale, a universal bound is given on the expected $p$-variation of $X$ on the interval $[0,T_\lambda]$, where $T_\lambda=\inf\{t:|X_t-X_0|\ge\lambda\}$. The main tool is Doob's classical upcrossing inequality
Comment: For an extension see
1319. These properties are used in T.~Lyons' pathwise theory of stochastic differential equations; see his long article in
Rev. Math. Iberoamericana 14, 1998
Keywords: $p$-variation,
UpcrossingsNature: Original Retrieve article from Numdam
XIII: 19, 233-237, LNM 721 (1979)
STRICKER, Christophe
Sur la $p$-variation des surmartingales (
Martingale theory)
The method of the preceding paper of Bruneau
1318 is extended to all right-continuous semimartingales
Keywords: $p$-variation,
UpcrossingsNature: Original Retrieve article from Numdam
XIII: 20, 238-239, LNM 721 (1979)
STRICKER, Christophe
Une remarque sur l'exposé précédent (
Martingale theory)
A few comments are added to the preceding paper
1319, concerning in particular its relationship with results of Lépingle,
Zeit. für W-Theorie, 36, 1976
Keywords: $p$-variation,
UpcrossingsNature: Original Retrieve article from Numdam
XIII: 21, 240-249, LNM 721 (1979)
MEYER, Paul-André
Représentations multiplicatives de sousmartingales, d'après Azéma (
Martingale theory)
The problem of the multiplicative decomposition of a positive supermartingale $Y$ is relatively easy, but the similar problem for a positive submartingale (find a previsible decreasing process $(C_t)$ such that $(C_tY_t)$ is a martingale) is plagued by the zeros of $Y$. An important idea of Azéma (
Zeit. für W-Theorie, 45, 1978) is the introduction of a
multiplicative system as a two-parameter process $C_{st}$ taking values in $[0,1]$, defined for $s\le t$, such that $C_{tt}=1$, $C_{st}C_{tu}=C_{su}$ for $s\le t\le u$, decreasing and previsible in $t$ for fixed $s$, such that $E[C_{st}Y_t\,|\,{\cal F}_s]=Y_s$ for $s<t$. Then for fixed $s$ the process $(C_{st}Y_t)$ turns out to be a right-continuous martingale on $[s,\infty[$, and what we have done amounts to pasting together all the multiplicative decompositions on zero-free intervals. Existence (and uniqueness of multiplicative systems are proved, though the uniqueness result is slightly different from Azéma's
Keywords: Multiplicative decompositionNature: Exposition,
Original additions Retrieve article from Numdam
XIII: 22, 250-252, LNM 721 (1979)
CHOU, Ching Sung
Caractérisation d'une classe de semimartingales (
Martingale theory,
Stochastic calculus)
The class of semimartingales $X$ such that the stochastic integral $J\,
.\,X$ is a martingale for some nowhere vanishing previsible process $J$ is a natural class of martingale-like processes. Local martingales are exactly the members of this class which are special semimartingales
Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997)
Keywords: Local martingales,
Stochastic integralsNature: Original Retrieve article from Numdam
XIII: 24, 260-280, LNM 721 (1979)
ÉMERY, Michel
Une topologie sur l'espace des semimartingales (
General theory of processes,
Stochastic calculus)
The stability theory for stochastic differential equations was developed independently by Émery (
Zeit. für W-Theorie, 41, 1978) and Protter (same journal,
44, 1978). However, these results were stated in the language of convergent subsequences instead of true topological results. Here a linear topology (like convergence in probability: metrizable, complete, not locally convex) is defined on the space of semimartingales. Side results concern the Banach spaces $H^p$ and $S^p$ of semimartingales. Several useful continuity properties are proved
Comment: This topology has become a standard tool. For its main application, see the next paper
1325Keywords: Semimartingales,
Spaces of semimartingalesNature: Original Retrieve article from Numdam
XIII: 25, 281-293, LNM 721 (1979)
ÉMERY, Michel
Équations différentielles stochastiques lipschitziennes~: étude de la stabilité (
Stochastic calculus)
This is the main application of the topologies on processes and semimartingales introduced in
1324. Using a very general definition of stochastic differential equations turns out to make the proof much simpler, and the existence and uniqueness of solutions of such equations is proved anew before the stability problem is discussed. Useful inequalities on stochastic integration are proved, and used as technical tools
Comment: For all of this subject, the book of Protter
Stochastic Integration and Differential Equations, Springer 1989, is a useful reference
Keywords: Stochastic differential equations,
StabilityNature: Original Retrieve article from Numdam
XIII: 26, 294-306, LNM 721 (1979)
BONAMI, Aline;
LÉPINGLE, Dominique
Fonction maximale et variation quadratique des martingales en présence d'un poids (
Martingale theory)
Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $
P$ two functionals of a $
P$-martingale---remains true, possibly with new constants, when $
P$ is replaced by an equivalent law $Z.
P$. To this order, the ``weight'' $Z$ must satisfy special conditions, among which a probabilistic version of Muckenhoupt's (1972) $(A_p)$ condition and a condition of multiplicative boundedness on the jumps of the martingale $E[Z\,|\,{\cal F}_t]$. This volume contains three papers on weighted norms inequalities,
1326,
1327,
1328, with considerable overlap. Here the main topic is the weighted-norm extension of the Burkholder-Gundy inequalities
Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance
Keywords: Weighted norm inequalities,
Burkholder inequalitiesNature: Original Retrieve article from Numdam
XIII: 27, 307-312, LNM 721 (1979)
IZUMISAWA, Masataka;
SEKIGUCHI, Takesi
Weighted norm inequalities for martingales (
Martingale theory)
See the review of
1326. The topic is the same, though the proof is different
Comment: See the paper by Kazamaki-Izumisawa in
Tôhoku Math. J. 29, 1977. For a modern reference see also Kazamaki,
Continuous Exponential Martingales and $\,BMO$, LNM. 1579, 1994
Keywords: Weighted norm inequalities,
Burkholder inequalitiesNature: Original Retrieve article from Numdam
XIII: 28, 313-331, LNM 721 (1979)
DOLÉANS-DADE, Catherine;
MEYER, Paul-André
Inégalités de normes avec poids (
Martingale theory)
See the review of
1326. This is a rather systematic exposition of the subject in the frame of martingale theory
Comment: An exponent $1/\lambda$ is missing in formula (4), p.315
Keywords: Weighted norm inequalitiesNature: Exposition,
Original additions Retrieve article from Numdam
XIII: 29, 332-359, LNM 721 (1979)
JEULIN, Thierry;
YOR, Marc
Inégalité de Hardy, semimartingales, et faux-amis (
Martingale theory,
General theory of processes)
The main purpose of this paper is to warn against ``obvious'' statements which are in fact false. Let $({\cal G}_t)$ be an enlargement of $({\cal F}_t)$. Assume that ${\cal F}$ has the previsible representation property with respect to a martingale $X$ which is a ${\cal G}$-semimartingale. Then it does not follow that every ${\cal F}$-martingale $Y$ is a ${\cal G}$-semimartingale. Also, even if $Y$ is a ${\cal G}$-semimartingale, its ${\cal G}$-compensator may have bad absolute continuity properties. The counterexample to the first statement involves a detailed study of the initial enlargement of the filtration of Brownian motion $(B_t)_{t\le 1}$ by the random variable $B_1$, which transforms it into the Brownian bridge, a semimartingale. Then the stochastic integrals with respect to $B$ which are ${\cal G}$-semimartingales are completely described, and this is the place where the classical Hardy inequality appears
Keywords: Hardy's inequality,
Previsible representationNature: Original Retrieve article from Numdam
XIII: 30, 360-370, LNM 721 (1979)
JEULIN, Thierry;
YOR, Marc
Sur l'expression de la dualité entre $H^1$ et $BMO$ (
Martingale theory)
The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$
Comment: On the same topic see
1518Keywords: $BMO$,
$H^1$ space,
Hardy spacesNature: Original Retrieve article from Numdam
XIII: 31, 371-377, LNM 721 (1979)
DELLACHERIE, Claude
Inégalités de convexité pour les processus croissants et les sousmartingales (
Martingale theory)
Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,
Probabilités et Potentiels B, Chapter VI
Keywords: Martingale inequalities,
Convex functionsNature: Exposition,
Original proofs Retrieve article from Numdam
XIII: 32, 378-384, LNM 721 (1979)
SZPIRGLAS, Jacques;
MAZZIOTTO, Gérald
Théorème de séparation dans le problème d'arrêt optimal (
General theory of processes)
Let $({\cal G}_t)$ be an enlargement of a filtration $({\cal F}_t)$ with the property that for every $t$, if $X$ is ${\cal G}_t$-measurable, then $E[X\,|\,{\cal F}_t]=E[X\,|\,{\cal F}_\infty]$. Then if $(X_t)$ is a ${\cal F}$-optional process, its Snell envelope is the same in both filtrations. Applications are given to filtering theory
Keywords: Optimal stopping,
Snell's envelope,
Filtering theoryNature: Original Retrieve article from Numdam
XIII: 33, 385-399, LNM 721 (1979)
LE JAN, Yves
Martingales et changement de temps (
Martingale theory,
Markov processes)
The first part of the paper concerns changes of time by a continuous (not strictly increasing) process, with a detailed computation, for instance, of the continuous martingale part of a time-changed martingale. This is a useful addition to
1108 and
1109. The second part is an application to classical potential theory: the martingale is a harmonic function along Brownian motion in a domain, stopped at the boundary; the change of time is defined by a boundary local time. Then the time-changed Brownian motion is a Markov process on the boundary, the time-changed martingale is purely discontinuous, and the computation of its quadratic norm leads to the Douglas formula, which expresses the Dirichlet integral of the harmonic function by a quadratic double integral of its restriction to the boundary
Keywords: Changes of time,
Energy,
Douglas formulaNature: Original Retrieve article from Numdam
XIII: 34, 400-406, LNM 721 (1979)
YOR, Marc
Quelques épilogues (
General theory of processes,
Martingale theory,
Stochastic calculus)
This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$
Keywords: Local time,
Enlargement of filtrations,
$H^1$ space,
Hardy spaces,
$BMO$Nature: Original Retrieve article from Numdam
XIII: 35, 407-426, LNM 721 (1979)
YOR, Marc
En cherchant une définition naturelle des intégrales stochastiques optionnelles (
Stochastic calculus)
While the stochastic integral of a previsible process is a very natural object, the optional (compensated) stochastic integral is somewhat puzzling: it concerns martingales only, and depends on the probability law. This paper sketches a ``pedagogical'' approach, using a version of Fefferman's inequality for thin processes to characterize those thin processes which are jump processes of local martingales. The results of
1121,
1129 are easily recovered. Then an attempt is made to extend the optional integral to semimartingales
Keywords: Optional stochastic integrals,
Fefferman inequalityNature: Original Retrieve article from Numdam
XIII: 36, 427-440, LNM 721 (1979)
YOR, Marc
Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (
Brownian motion)
The problem is to study the filtration generated by real valued stochastic integrals $Y=\int_0^t(AX_s, dX_s)$, where $X$ is a $n$-dimensional Brownian motion, $A$ is a $n\times n$-matrix, and $(\,,\,)$ is the scalar product. If $A$ is the identity matrix we thus get (squares of) Bessel processes. If $A$ is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of $A$. In particular, this dimension is $1$ if and only if the matrix is equivalent to $cI_r$, a diagonal with $r$ ones and $n-r$ zeros. This is also (even if the symmetry assumption is omitted) the only case where $Y$ has the previsible representation property
Comment: Additional results on the same subject appear in
1545 and in Malric
Ann. Inst. H. Poincaré 26 (1990)
Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XIII: 37, 441-442, LNM 721 (1979)
CHOU, Ching Sung
Démonstration simple d'un résultat sur le temps local (
Stochastic calculus)
It follows from Ito's formula that the positive parts of those jumps of a semimartingale $X$ that originate below $0$ are summable. A direct proof is given of this fact
Comment: Though the idea is essentially correct, an embarrassing mistake is corrected as
1429Keywords: Local times,
Semimartingales,
JumpsNature: Original Retrieve article from Numdam
XIII: 38, 443-452, LNM 721 (1979)
EL KAROUI, Nicole
Temps local et balayage des semimartingales (
General theory of processes)
This paper is the first one in a series of reports on the balayage of semimartingales, and the following description is common to all of them. \par Let $H$ be a right-closed optional set, and let $g_t=\sup\{s<t, s\in H\}$ and $D_t=\inf\{s>t,s\in H\}$. Put $L=g_{\infty}$. Let also $G$ be the set of all left-endpoints of intervals contiguous to $H$, i.e., of all points $g_t$ for $t\notin H$. For simplicity we assume here that $D_0=0$ and that $H=\{X=0\}$, where $X$ is a semimartingale with decomposition $X=M+V$, though for a few results (including the balayage formula itself) it is sufficient that $X=0$ on $H$. \par One of the starting points of this paper is the
balayage formula (see Azéma-Yor, introduction to
Temps Locaux ,
Astérisque ,
52-53): if $Z$ is a locally bounded previsible process, then $$Z_{g_t}X_t=\int_0^t Z_{g_s}dX_s$$ and therefore $Y_t=Z_{g_t}X_t$ is a semimartingale. The main problem of the series of reports is: what can be said if $Z$ is not previsible, but optional, or even progressive?\par This particular paper is devoted to the study of the non-adapted process $$K_t=\sum_{g\in G,g\le t } (M_{D_g}-M_g)$$ which turns out to have finite variation
Comment: This paper is completed by
1357Keywords: Local times,
Balayage,
Balayage formulaNature: Original Retrieve article from Numdam
XIII: 39, 453-471, LNM 721 (1979)
YOR, Marc
Sur le balayage des semi-martingales continues (
General theory of processes)
For the general notation, see
1338. This paper is independent from the preceding one
1338, and some overlap occurs. The balayage formula is extended to processes $Z$ which are not locally bounded, and the local time of the semimartingale $Y$ is computed. The class of continuous semimartingales $X$ with canonical decomposition $X=M+V$ such that $dV$ is carried by $H=\{X=0\}$ is introduced and studied. It turns out to be an important class, closely related to ``relative martingales'' (Azéma, Meyer and Yor
2623). A number of results are given, too technical to be stated here. Stopping previsible, optional and progressive processes at the last exit time $L$ from $H$ leads to three $\sigma$-fields, ${\cal F}_L^p$, ${\cal F}_L^o$, ${\cal F}_L^{\pi}$, and it was considered surprising that the last two could be different (see
1240). Here it is shown that if $X$ is a continuous uniformly integrable martingale with $X_0=0$, $E[X_{\infty}|{\cal F}_L^o]=0\neq E[X_{\infty}|{\cal F}_L^{\pi}]$
Comment: See
1357Keywords: Local times,
Balayage,
Balayage formulaNature: Original Retrieve article from Numdam
XIII: 40, 472-477, LNM 721 (1979)
STRICKER, Christophe
Semimartingales et valeur absolue (
General theory of processes)
For the general notation, see
1338. A result of Yoeurp that absolute values preserves quasimartingales is extended: convex functions satisfying a Lipschitz condition operate on quasimartingales. For $p\ge1$, $X\in H^p$ implies $|X|^p\in H^1$. Then it is shown that for a continuous adapted process $X$, it is equivalent to say that $X$ and $|X|$ are quasimartingales (or semimartingales). Then comes a result related to the main problem of this series: with the general notations above, if $X$ is assumed to be a quasimartingale such that $X_{D_t}=0$ for all $t$, if the process $Z$ is progressive and bounded, then the process $Z_{g_t}X_t$ is a quasimartingale
Comment: A complement is given in the next paper
1341. See also
1351Keywords: Balayage,
QuasimartingalesNature: Original Retrieve article from Numdam
XIII: 41, 478-487, LNM 721 (1979)
MEYER, Paul-André;
STRICKER, Christophe;
YOR, Marc
Sur une formule de la théorie du balayage (
General theory of processes)
For the notation, see the review of
1340. It is shown here that under the same hypotheses, the semimartingale $Z_{g_t}X_t$ is a sum of three terms: the stochastic integral $\int_0^t \zeta_s dX_s$, where $\zeta$ is the previsible projection of $Z$, an explicit sum of jumps involving $Z-\zeta$, and a mysterious continuous process with finite variation $(R_t)$ such that $dR_t$ is carried by $H$, equal to $0$ if $Z$ was optional
Comment: See
1351,
1357Keywords: Balayage,
Balayage formulaNature: Original Retrieve article from Numdam
XIII: 42, 488-489, LNM 721 (1979)
MEYER, Paul-André
Construction de semimartingales s'annulant sur un ensemble donné (
General theory of processes)
The title states exactly the subject of this short report, whose conclusion is: in the Brownian filtration, every closed optional set is the set of zeros of a continuous semimartingale
Keywords: SemimartingalesNature: Original Retrieve article from Numdam
XIII: 43, 490-494, LNM 721 (1979)
WILLIAMS, David
Conditional excursion theory (
Brownian motion,
Markov processes)
To be completed
Keywords: ExcursionsNature: Original Retrieve article from Numdam
XIII: 44, 495-520, LNM 721 (1979)
BISMUT, Jean-Michel
Problèmes à frontière libre et arbres de mesures (
Miscellanea,
Markov processes)
An optimization problem is discussed, in which one is free to choose at any time among three different transition semi-groups
Keywords: Control theoryNature: Original Retrieve article from Numdam
XIII: 46, 533-547, LNM 721 (1979)
NANOPOULOS, Photius
Mesures de probabilités sur les entiers et ensembles progressions (
Miscellanea)
To be completed
Nature: Original Retrieve article from Numdam
XIII: 47, 548-556, LNM 721 (1979)
FUJISAKI, Masatoshi
On the uniqueness of optimal controls (
Miscellanea)
``In section 1 we can give simple criteria for the uniqueness of the optimal controls whose existence is proved by Ikeda-Watanabe in the completely observable case,
Osaka Math. J.,
14, 1977. In section 2 we consider the same problem in the partially observable case.'' (From the author's summary)
Keywords: Control theoryNature: Original Retrieve article from Numdam
XIII: 48, 557-569, LNM 721 (1979)
CARMONA, René
Processus de diffusion gouverné par la forme de Dirichlet de l'opérateur de Schrödinger (
Diffusion theory)
Standard conditions on the potential $V$ imply that the Schrödinger operator $-(1/2)ėlta+V$ (when suitably interpreted) is essentially self-adjoint on $L^2(
R^n,dx)$. Assume it has a ground state $\psi$. Then transferring everything on the Hilbert space $L^2(\mu)$ where $\mu$ has the density $\psi^2$ the operator becomes formally $Df=(-1/2)ėlta f + \nabla h.\nabla f$ where $h=-log\psi$. A problem which has aroused some excitement ( due in part to Nelson's ``stochastic mechanics'') was to construct true diffusions governed by this generator, whose meaning is not even clearly defined unless $\psi$ satisfies regularity conditions, unnatural in this problem. Here a reasonable positive answer is given
Comment: This problem, though difficult, is but the simplest case in Nelson's theory. In this seminar, see
1901,
1902,
2019. Seemingly definitive results on this subject are due to E.~Carlen,
Comm. Math. Phys.,
94, 1984. A recent reference is Aebi,
Schrödinger Diffusion Processes, Birkhäuser 1995
Keywords: Nelson's stochastic mechanics,
Schrödinger operatorsNature: Original Retrieve article from Numdam
XIII: 49, 570-573, LNM 721 (1979)
CARMONA, René
Opérateur de Schrödinger à résolvante compacte (
Miscellanea)
A sufficient condition for a Schrödinger operator $(-1/2)ėlta+V$ to have a compact resolvent is proved, using standard properties of Brownian paths
Keywords: Schrödinger operatorsNature: Original Retrieve article from Numdam
XIII: 50, 574-609, LNM 721 (1979)
JEULIN, Thierry
Grossissement d'une filtration et applications (
General theory of processes,
Markov processes)
This is a sequel to the papers
1209 and
1211, giving mostly applications of the theory of enlargements (turning a honest time $L$ into a stopping time) to Markov processes. The paper begins with a computation of conditional expectations relative to ${\cal F}_{L-}$, ${\cal F}_{L}$, ${\cal F}_{L+}$. This result is applied to coterminal times of a Markov process. Again a section is devoted to a general computation on two successive enlargements, which is shown to imply (with some work) Williams' well-known decomposition of Brownian paths
Keywords: Enlargement of filtrations,
Williams decompositionNature: Original Retrieve article from Numdam
XIII: 51, 610-610, LNM 721 (1979)
STRICKER, Christophe
Encore une remarque sur la ``formule de balayage'' (
General theory of processes)
A slight extension of
1341Keywords: BalayageNature: Original Retrieve article from Numdam
XIII: 53, 614-619, LNM 721 (1979)
YOEURP, Chantha
Solution explicite de l'équation $Z_t=1+\int_0^t |Z_{s-}|\,dX_s$ (
Stochastic calculus)
The title describes completely the paper
Keywords: Stochastic differential equationsNature: Original Retrieve article from Numdam
XIII: 56, 625-633, LNM 721 (1979)
AZÉMA, Jacques;
YOR, Marc
Le problème de Skorokhod~: compléments à l'exposé précédent (
Brownian motion)
What the title calls ``the preceding talk'' is
1306. The method is extended to (centered) measures possessing a moment of order one instead of two, preserving the uniform integrability of the stopped martingale
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XIII: 57, 634-641, LNM 721 (1979)
EL KAROUI, Nicole
A propos de la formule d'Azéma-Yor (
General theory of processes)
For the problem and notation, see the review of
1340. The problem is completely solved here, the process $Z_{g_t}X_t$ being represented as the sum of $\int_0^t Z_{g_s}dX_s$ interpreted in a generalized sense ($Z$ being progressive!) and a remainder which can be explicitly written (using optional dual projections of non-adapted processes)
Comment: This paper ends happily the whole series of papers on balayage in this volume
Keywords: Balayage,
Balayage formulaNature: Original Retrieve article from Numdam
XIII: 58, 642-645, LNM 721 (1979)
MAISONNEUVE, Bernard
Martingales de valeur absolue donnée, d'après Protter-Sharpe (
Martingale theory)
The main difficulty of Gilat's theorem (every positive submartingale $X$ can be interpreted as the absolute value of a martingale, in a suitably enlarged filtration) is due to the zeros of $X$. In the strictly positive case a simple proof was given by Protter and Sharpe (
Ann. Prob.,
7, 1979). This proof is further simplified and slightly generalized
Comment: See also
1407Keywords: Gilat's theoremNature: Exposition,
Original additions Retrieve article from Numdam
XIII: 59, 646-646, LNM 721 (1979)
BARLOW, Martin T.
On the left endpoints of Brownian excursions (
Brownian motion,
Excursion theory)
It is shown that no expansion of the Brownian filtration can be found such that $B_t$ remains a semimartingale, and the set of left endpoints of Brownian excursions becomes optional
Keywords: Progressive setsNature: Original Retrieve article from Numdam
XIV: 01, 1-16, LNM 784 (1980)
HEINKEL, Bernard
Deux exemples d'utilisation de mesures majorantes (
Gaussian processes)
From the introduction: ``our purpose is to study in detail two examples to which the method of majorizing measures, but not the entropy method, can be applied''
Nature: Original Retrieve article from Numdam
XIV: 03, 18-25, LNM 784 (1980)
CAIROLI, Renzo
Sur l'extension de la définition d'intégrale stochastique (
Several parameter processes)
A result of Wong-Zakai (
Ann. Prob. 5, 1977) extending the definition of the two kinds of stochastic integrals relative to the Brownian sheet is generalized to cover the case of stochastic integration relative to martingales, or strong martingales
Comment: A note at the end of the paper suggests some improvements
Keywords: Stochastic integrals,
Brownian sheetNature: Original Retrieve article from Numdam
XIV: 04, 26-48, LNM 784 (1980)
LENGLART, Érik;
LÉPINGLE, Dominique;
PRATELLI, Maurizio
Présentation unifiée de certaines inégalités de la théorie des martingales (
Martingale theory)
This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes
Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,
Probabilités et Potentiel E ; see also
1621. A striking example of the power of these methods is Barlow-Yor, {\sl Jour. Funct. Anal.}
49,1982
Keywords: Moderate convex functions,
Inequalities,
Martingale inequalities,
Burkholder inequalities,
Good lambda inequalities,
Domination inequalitiesNature: Original Retrieve article from Numdam
XIV: 05, 49-52, LNM 784 (1980)
LENGLART, Érik
Appendice à l'exposé précédent~: inégalités de semimartingales (
Martingale theory,
Stochastic calculus)
This paper contains several applications of the methods of
1404 to the case of semimartingales instead of martingales
Keywords: Inequalities,
SemimartingalesNature: Original Retrieve article from Numdam
XIV: 06, 53-61, LNM 784 (1980)
AZÉMA, Jacques;
GUNDY, Richard F.;
YOR, Marc
Sur l'intégrabilité uniforme des martingales exponentielles (
Martingale theory)
The main result of this paper is the following: Let $X$ be a martingale which is continuous and bounded in $L^1$ (both conditions are essential). Then $X$ is uniformly integrable if and only if $tP\{X^{*}>t\}$ or equivalently $tP\{S(X)>t\}$ tend to $0$ as $t\rightarrow\infty$, where $S(X)$ is the usual square function. The methods (using a good lambda inequality) are close to
1404Comment: Generalized by Takaoka
3313Keywords: Exponential martingales,
Continuous martingalesNature: Original Retrieve article from Numdam
XIV: 07, 62-75, LNM 784 (1980)
BARLOW, Martin T.;
YOR, Marc
Sur la construction d'une martingale continue de valeur absolue donnée (
Martingale theory)
This paper consists of two notes on Gilat's theorem (
Ann. Prob. 5, 1977, See also
1358). The problem consists in constructing, given a continuous positive submartingale $Y$, a
continuous martingale $X$ (possibly on a different space) such that $|X|$ has the same law as $Y$. Let $A$ be the increasing process associated with $Y$; it is necessary for the existence of $X$ that $dA$ should be carried by $\{Y=0\}$. This is shown by the first note (Yor's) to be also sufficient---more precisely, in this case the solutions of Gilat's problem are all continuous. The second note (Barlow's) shows how to construct a Gilat martingale by ``putting a random $\pm$ sign in front of each excursion of $Y$'', a simple intuitive idea and a delicate proof
Keywords: Gilat's theoremNature: Original Retrieve article from Numdam
XIV: 08, 76-101, LNM 784 (1980)
SHARPE, Michael J.
Local times and singularities of continuous local martingales (
Martingale theory)
This paper studies continuous local martingales $(M_t)$ in the open interval $]0,\infty[$. After recalling a few useful results on local martingales, the author proves that the sample paths a.s., either have a limit (possibly $\pm\infty$) at $t=0$, or oscillate over the whole interval $]-\infty,\infty[$ (this is due to Walsh
1133, but the proof here does not use conformal martingales). Then the quadratic variation and local time of $M$ are defined as random measures which may explode near $0$, and it is shown that non-explosion of the quadratic variation (of the local time) measure characterizes the sample paths which have a finite limit (a limit) at $0$. The results are extended in part to local martingale increment processes, which are shown to be stochastic integrals with respect to true local martingales, of previsible processes which are not integrable near $0$
Comment: See Calais-Genin
1717Keywords: Local times,
Local martingales,
Semimartingales in an open intervalNature: Original Retrieve article from Numdam
XIV: 09, 102-103, LNM 784 (1980)
MEYER, Paul-André
Sur un résultat de L. Schwartz (
Martingale theory)
the following definition of a semimartingale $X$ in a random open set $A$ is due to L. Schwartz (
Semimartingales dans les variétés..., Lecture Notes in M.
780): $A$ can be represented as a countable union of random open sets $A_n$, and for each $n$ there exists an ordinary semimartingale $Y_n$ such $X=Y_n$ on $A_n$. It is shown that if $K\subset A$ is a compact optional set, then there exists an ordinary semimartingale $Y$ such that $X=Y$ on $K$
Comment: The results are extended in Meyer-Stricker
Stochastic Analysis and Applications, part B, Advances in M. Supplementary Studies, 1981
Keywords: Semimartingales in a random open setNature: Exposition,
Original additions Retrieve article from Numdam
XIV: 10, 104-111, LNM 784 (1980)
STRICKER, Christophe
Prolongement des semi-martingales (
Stochastic calculus)
The problem consists in characterizing semimartingales on $]0,\infty[$ which can be ``closed at infinity'', and the similar problem at $0$. The criteria are similar to the Vitali-Hahn-Saks theorem and involve convergence in probability of suitable stochastic integrals. The proof rests on a functional analytic result of Maurey-Pisier
Keywords: Semimartingales,
Semimartingales in an open intervalNature: Original Retrieve article from Numdam
XIV: 11, 112-115, LNM 784 (1980)
STRICKER, Christophe
Projection optionnelle des semi-martingales (
Stochastic calculus)
Let $({\cal G}_t)$ be a subfiltration of $({\cal F}_t)$. Since the optional projection on $({\cal G}_t)$ of a ${\cal F}$-martingale is a ${\cal G}$-martingale, and the projection of an increasing process a ${\cal G}$-submartingale, projections of ${\cal F}$-semimartingales ``should be'' ${\cal G}$-semimartingales. This is true for quasimartingales, but false in general
Comment: The main results on subfiltrations are proved by Stricker in
Zeit. für W-Theorie, 39, 1977
Keywords: Semimartingales,
Projection theoremsNature: Original Retrieve article from Numdam
XIV: 12, 116-117, LNM 784 (1980)
CHOU, Ching Sung
Une caractérisation des semimartingales spéciales (
Stochastic calculus)
This is a useful addition to the next paper
1413: a semimartingale can be ``controlled'' (in the sense of Métivier-Pellaumail) by a locally integrable increasing process if and only if it is special
Comment: See also
1352Keywords: Semimartingales,
Métivier-Pellaumail inequality,
Special semimartingalesNature: Original Retrieve article from Numdam
XIV: 13, 118-124, LNM 784 (1980)
ÉMERY, Michel
Équations différentielles stochastiques. La méthode de Métivier-Pellaumail (
Stochastic calculus)
Métivier-Pellaumail introduced the idea of an increasing process $(A_t)$ controlling a semimartingale $X$ as the property $$E[\,(sup_{t<T} \int_0^t H_s dX_s)^2\,] \le E[\,A_{T-}\,\int_0^{T-} H_s^2 dA_s\,]$$ for all stopping times $T$ and bounded previsible processes $(H_t)$. For a proof see
1414. Métivier-Pellaumail used this inequality to develop the theory of stochastic differential equations (including stability) without localization and pasting together at jump times. Here their method is applied to the topology of semimartingales
Comment: See
1352. A general reference on the Métivier-Pellaumail method can be found in their book
Stochastic Integration, Academic Press 1980. See also He-Wang-Yan,
Semimartingale Theory and Stochastic Calculus, CRC Press 1992
Keywords: Semimartingales,
Spaces of semimartingales,
Stochastic differential equations,
Doob's inequality,
Métivier-Pellaumail inequalityNature: Original Retrieve article from Numdam
XIV: 15, 128-139, LNM 784 (1980)
CHOU, Ching Sung;
MEYER, Paul-André;
STRICKER, Christophe
Sur l'intégrale stochastique de processus prévisibles non bornés (
Stochastic calculus)
The standard theory of stochastic integration deals with locally bounded previsible processes. The natural definition of the stochastic integral $H.X$ of a previsible process $H$ w.r.t. a semimartingale $X$ consists in assuming the existence of some decomposition $X=M+A$ such that $H.M$ exists in the martingale sense, and $H.A$ in the Stieltjes sense, and then defining $H.X$ as their sum. This turns out to be a very awkward definition. It is shown here to be equivalent to the following one: truncating $H$ at $n$, the standard stochastic integrals $H_n.X$ converge in the topology of semimartingales. This is clearly invariant under changes of law. A counterexample shows that integrability may be lost if the filtration is enlarged
Comment: See also
1417. This is a synthesis of earlier work, much of which is due to Jacod,
Calcul Stochastique et Problèmes de Martingales, Lect. Notes in M. 714. The contents of this paper appeared in book form in Dellacherie-Meyer,
Probabilités et Potentiel B, Chap. VIII, \S3. An equivalent definition is given by L. Schwartz in
1530, using the idea of ``formal semimartingales''. For further steps in the same direction, see Stricker
1533Keywords: Stochastic integralsNature: Exposition,
Original additions Retrieve article from Numdam
XIV: 16, 140-147, LNM 784 (1980)
ÉMERY, Michel
Métrisabilité de quelques espaces de processus aléatoires (
General theory of processes,
Stochastic calculus)
As a sequel to the main work of
1324 on the topology of semimartingales, several spaces of processes defined by localization (or prelocalization) of standard spaces of martingales or processes of bounded variation are studied here, and shown to be metrizable and complete
Keywords: Spaces of semimartingalesNature: Original Retrieve article from Numdam
XIV: 17, 148-151, LNM 784 (1980)
YAN, Jia-An
Remarques sur l'intégrale stochastique de processus non bornés (
Stochastic calculus)
It is shown how to develop the integration theory of unbounded previsible processes (due to Jacod
1126), starting from the elementary definition considered ``awkward'' in
1415Comment: Another approach to those integrals is due to L. Schwartz, in his article
1530 on formal semimartingales
Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XIV: 18, 152-160, LNM 784 (1980)
ÉMERY, Michel
Compensation de processus à variation finie non localement intégrables (
General theory of processes,
Stochastic calculus)
First an example is given of a local martingale $M$ and an unbounded previsible process $H$ such that $H.M$ exists in the sense of
1126 and
1415, but is not a local martingale. This leads to a natural enlargement of the class of local martingales, which turns out to be the same suggested by Chou in
1322 under the name of class $(\Sigma_m)$. Once the class has been so extended, the operation of previsible compensation can be extended to a class of processes with finite variation, but not locally integrable variation, and the class of special semimartingales can be also enlarged
Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997). Using the language of L. Schwartz
1530, it is the intersection of the set of (usual) semimartingales with the set of formal martingales
Keywords: Local martingales,
Stochastic integrals,
CompensatorsNature: Original Retrieve article from Numdam
XIV: 19, 161-172, LNM 784 (1980)
JACOD, Jean
Intégrales stochastiques par rapport à une semi-martingale vectorielle et changements de filtration (
Stochastic calculus,
General theory of processes)
Given a square integrable vector martingale $M$ and a previsible vector process $H$, the conditions implying the existence of the (scalar valued) stochastic integral $H.M$ are less restrictive than the existence of the ``componentwise'' stochastic integral, unless the components of $M$ are orthogonal (this result was due to Galtchouk, 1975). The theory of vector stochastic integrals, though parallel to the scalar theory, requires a careful theory given in this paper
Comment: Another approach, yielding an equivalent definition, is followed by L. Schwartz in his article
1530 on formal semimartingales
Keywords: Semimartingales,
Stochastic integralsNature: Original Retrieve article from Numdam
XIV: 21, 189-199, LNM 784 (1980)
YOR, Marc
Application d'un lemme de Jeulin au grossissement de la filtration brownienne (
General theory of processes,
Brownian motion)
The problem considered here is the smallest enlargement of the Brownian filtration for which the process $\int_t^\infty B_s\mu(ds)$ is adapted, $\mu$ being a probability measure with a finite first moment
Comment: Note the misprint ${\cal G}$-martingale instead of ${\cal G}$-semimartingale in the statement of condition (H')
Keywords: Enlargement of filtrationsNature: Original Retrieve article from Numdam
XIV: 22, 200-204, LNM 784 (1980)
AUERHAN, J.;
LÉPINGLE, Dominique;
YOR, Marc
Construction d'une martingale réelle continue de filtration naturelle donnée (
General theory of processes)
It is proved in
1123 that, under a mere separability assumption, any filtration is the natural filtration of some bounded left-continuous increasing process $(A_t)$. If the filtration contains a Brownian motion $(B_t)$, then it is also the natural filtration of the continuous martingale $\int_0^t A_sdB_s$. Therefore, the natural filtration of (finitely or countably) many independent Brownian motions is generated by a single continuous martingale. Explicit constructions are discussed
Nature: Original Retrieve article from Numdam
XIV: 23, 205-208, LNM 784 (1980)
SEYNOU, Aboubakary
Sur la compatibilité temporelle d'une tribu et d'une filtration discrète (
General theory of processes)
Let us say that a $\sigma$-field ${\cal G}$ is embedded into a filtration $({\cal H}_n)$ if ${\cal G}= {\cal H}_T$ for some ${\cal H}$-stopping time $T$, that a filtration $({\cal F}_n)$ is embedded into $({\cal H}_n)$ if ${\cal F}_S$ can be embedded into $({\cal H}_n)$ for every ${\cal F}$-stopping time $S$, and finally that ${\cal G}$ and ${\cal F}$ or $({\cal F}_n)$ are compatible if they can be embedded into one single filtration $({\cal H}_n)$. Then the question investigated is whether the compatibility of ${\cal G}$ and the individual $\sigma$-fields ${\cal F}_n$ implies that of ${\cal G}$ and the whole filtration $({\cal F}_n)$
Comment: This problem arose from the spectral point of view on stochastic integration as in
1123Keywords: FiltrationsNature: Original Retrieve article from Numdam
XIV: 25, 220-222, LNM 784 (1980)
YAN, Jia-An
Caractérisation d'ensembles convexes de $L^1$ ou $H^1$ (
Stochastic calculus,
Functional analysis)
This is a new and simpler approach to the crucial functional analytic lemma in
1354 (the proof that semimartingales are the stochastic integrators in $L^0$). That is, given a convex set $K\subset L^1$ containing $0$, find a condition for the existence of $Z>0$ in $L^\infty$ such that $\sup_{X\in K}E[ZX]<\infty$. A similar result is discussed for $H^1$ instead of $L^1$
Comment: This lemma, instead of the original one, has proved very useful in mathematical finance
Keywords: Semimartingales,
Stochastic integrals,
Convex functionsNature: Original Retrieve article from Numdam
XIV: 26, 223-226, LNM 784 (1980)
YAN, Jia-An
Remarques sur certaines classes de semimartingales et sur les intégrales stochastiques optionnelles (
Stochastic calculus)
A class of semimartingales containing the special ones is introduced, which can be intrinsically decomposed into a continuous and a purely discontinuous part. These semimartingales have ``not too large totally inaccessible jumps''. In the second part of the paper, a non-compensated optional stochastic integral is defined, improving the results of Yor
1335Keywords: Semimartingales,
Optional stochastic integralsNature: Original Retrieve article from Numdam
XIV: 27, 227-248, LNM 784 (1980)
JACOD, Jean;
MÉMIN, Jean
Sur la convergence des semimartingales vers un processus à accroissements indépendants (
General theory of processes,
Stochastic calculus,
Martingale theory)
A method of Kabanov, Liptzer and Shiryaev is adapted to study the convergence of a sequence of semimartingales to a process with independent increments (to be completed)
Keywords: Convergence in law,
TightnessNature: Original Retrieve article from Numdam
XIV: 28, 249-253, LNM 784 (1980)
YOEURP, Chantha
Sur la dérivation des intégrales stochastiques (
Stochastic calculus)
The following problem is discussed: under which conditions do ratios of the form $\int_t^{t+h} H_s\,dM_s/(M_{t+h}-M_t)$ converge to $H_t$ as $h\rightarrow 0$? It is shown that positive results due to Isaacson (
Ann. Math. Stat. 40, 1979) in the Brownian case fail in more general situations
Comment: See also
1529Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XIV: 31, 256-281, LNM 784 (1980)
FUJISAKI, Masatoshi
Contrôle stochastique continu et martingales (
General theory of processes)
To be completed
Keywords: Control theoryNature: Original Retrieve article from Numdam
XIV: 32, 282-304, LNM 784 (1980)
KUNITA, Hiroshi
On the representation of solutions of stochastic differential equations (
Stochastic calculus)
This paper concerns stochastic differential equations in the standard form $dY_t=\sum_i X_i(Y_t)\,dB^i(t)+X_0(Y_t)\,dt$ where the $B^i$ are independent Brownian motions, the stochastic integrals are in the Stratonovich sense, and $X_i,X_0$ have the geometric nature of vector fields. The problem is to find a deterministic (and smooth) machinery which, given the paths $B^i(.)$ will produce the path $Y(.)$. The complexity of this machinery reflects that of the Lie algebra generated by the vector fields. After a study of the commutative case, a paper of Yamato settled the case of a nilpotent Lie algebra, and the present paper deals with the solvable case. This line of thought led to the important and popular theory of flows of diffeomorphisms associated with a stochastic differential equation (see for instance Kunita's paper in
Stochastic Integrals, Lecture Notes in M. 851)
Comment: On a closely related subject, see the paper of Fliess and Norman-Cyrot,
1623Keywords: Stochastic differential equations,
Lie algebras,
Campbell-Hausdorff formulaNature: Original Retrieve article from Numdam
XIV: 33, 305-315, LNM 784 (1980)
YAN, Jia-An
Sur une équation différentielle stochastique générale (
Stochastic calculus)
The differential equation considered is of the form $X_t= \Phi(X)_t+\int_0^tF(X)_s\,dM_s$, where $M$ is a semimartingale, $\Phi$ maps adapted cadlag processes into themselves, and $F$ maps adapted cadlag process into previsible processes---not locally bounded, this is the main technical point. Some kind of Lipschitz condition being assumed, existence, uniqueness and stability are proved
Keywords: Stochastic differential equationsNature: Original Retrieve article from Numdam
XIV: 34, 316-317, LNM 784 (1980)
ÉMERY, Michel
Une propriété des temps prévisibles (
General theory of processes)
The idea is to prove that the general theory of processes on a random interval $[0,T[$, where $T$ is previsible, is essentially the same as on $[0,\infty[$. To this order, a continuous, strictly increasing adapted process $(A_t)$ is constructed, such that $A_0=0$, $A_T=1$
Keywords: Previsible timesNature: Original Retrieve article from Numdam
XIV: 35, 318-323, LNM 784 (1980)
ÉMERY, Michel
Annonçabilité des temps prévisibles. Deux contre-exemples (
General theory of processes)
It is shown that two standard results on previsible stopping times on a probability space, namely that every previsible time $T$ can be ``foretold'' by a strictly increasing sequence $T_n\uparrow T$, and that the $T_n$ can themselves be taken previsible, become false if exceptional sets of measure zero are not allowed
Keywords: Previsible timesNature: Original Retrieve article from Numdam
XIV: 36, 324-331, LNM 784 (1980)
BARLOW, Martin T.;
ROGERS, L.C.G.;
WILLIAMS, David
Wiener-Hopf factorization for matrices (
Markov processes)
Let $(X_t)$ be a continuous-time Markov chain with a finite state space $E$, and a transition semigroup $\exp(tQ)$. Consider the fluctuating additive functional $\phi_t=\int_0^t v(X_s)\,ds$ ($v$ is a function on $E$ which may assume negative values) and the corresponding change of time $\tau_t= \inf\{s:\phi_s>t\}$. The problem is to find the joint distribution of $\tau_t$ and $X(\tau_t)$. This is solved using martingale methods, and implies a purely algebraic result on the structure of the Q-matrix
Comment: A mistake is pointed out by the authors at the end of the paper, and is corrected in
1437Keywords: Wiener-Hopf factorizations,
Additive functionals,
Changes of time,
Markov chainsNature: Original Retrieve article from Numdam
XIV: 37, 332-342, LNM 784 (1980)
ROGERS, L.C.G.;
WILLIAMS, David
Time-substitution based on fluctuating additive functionals (Wiener-Hopf factorization for infinitesimal generators) (
Markov processes)
This is a first step towards the extension of
1436 to Markov processes with a general state space
Keywords: Wiener-Hopf factorizations,
Additive functionals,
Changes of timeNature: Original Retrieve article from Numdam
XIV: 38, 343-346, LNM 784 (1980)
YOR, Marc
Remarques sur une formule de Paul Lévy (
Brownian motion)
Given a two-dimensional Brownian motion $(X_t,Y_t)$, Lévy's area integral formula gives the characteristic function $E[\,\exp(iu\int_0^1 X_s\,dY_s-Y_s\,dX_s)\,\,|\,\, X_0=x, Y_0=y]$. A short proof of this formula is given, and it is shown how to deduce from it the apparently more general $E[\exp(iu\int_0^1 X_sdY_s+iv\int_0^1 Y_sdX_s)\,]$ computed by Berthuet
Keywords: Area integral formulaNature: Original Retrieve article from Numdam
XIV: 39, 347-356, LNM 784 (1980)
CHUNG, Kai Lai
On stopped Feynman-Kac functionals (
Markov processes,
Diffusion theory)
Let $(X_t)$ be a strong Markov process with continuous paths on the line, and let $\tau_b$ be the hitting time of the point $b$. It is assumed that $\tau_b$ is $P_a$-a.s. finite for all $a,b$. The purpose of the paper is to study the quantities $u(a,b)=E_a[\,\exp(\int_0^{\tau_b} q(X_s)\,ds)\,]$ where $q$ is bounded. Then (among other results) if $u(a,b)<\infty$ for all $a<b$, we have $u(a,b)\,u(b,a)\le 1$ for all $a,b$
Keywords: Hitting probabilitiesNature: Original Retrieve article from Numdam
XIV: 40, 357-391, LNM 784 (1980)
FALKNER, Neil
On Skorohod embedding in $n$-dimensional Brownian motion by means of natural stopping times (
Brownian motion,
Potential theory)
The problem discussed here is the Skorohod representation of a measure $\nu$ as the distribution of $B_T$, where $(B_t)$ is Brownian motion in $
R^n$ with the initial measure $\mu$, and $T$ is a
non-randomized stopping time. The conditions given are sufficient in all cases, necessary if $\mu$ does not charge polar sets
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XIV: 41, 392-396, LNM 784 (1980)
PIERRE, Michel
Le problème de Skorohod~: une remarque sur la démonstration d'Azéma-Yor (
Brownian motion)
This is an addition to
1306 and
1356, showing how the proof can be reduced to that of a regular case, where it becomes simpler
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XIV: 42, 397-409, LNM 784 (1980)
GETOOR, Ronald K.
Transience and recurrence of Markov processes (
Markov processes)
From the introduction: The purpose of this paper is to present an elementary exposition of some various conditions that have been used to define transience or recurrence of a Markov process... an elementary and unified discussion of these ideas may be worthwhile
Keywords: Recurrent Markov processesNature: Exposition,
Original additions Retrieve article from Numdam
XIV: 43, 410-417, LNM 784 (1980)
JACOD, Jean;
MAISONNEUVE, Bernard
Remarque sur les fonctionnelles additives non adaptées des processus de Markov (
Markov processes)
It occurs sometimes that a Markov process $(X_t)$ satisfies in a filtration ${\cal H}_t$ a Markov property of the form $E[f\circ \theta_t \,|\,{\cal H}_t]= E_{X_t}[f]$, where $f$ is not restricted to be ${\cal H}_t$-measurable. For instance, situations in renewal theory where one is given a Markov pair $(X_t,Y_t)$, and ${\cal H}_t$ describes the path of $X$ up to time $t$, and the whole path of $Y$. In such cases, the authors show that additive functionals which are previsible in the larger filtration are in fact previsible in the filtration of $X$ alone
Keywords: Additive functionalsNature: Original Retrieve article from Numdam
XIV: 44, 418-436, LNM 784 (1980)
RAO, Murali
A note on Revuz measure (
Markov processes,
Potential theory)
The problem is to weaken the hypotheses of Chung (
Ann. Inst. Fourier, 23, 1973) implying the representation of the equilibrium potential of a compact set as a Green potential. To this order, Revuz measure techniques are used, and interesting auxiliary results are proved concerning the Revuz measures of natural additive functionals of a Hunt process
Keywords: Revuz measures,
Additive functionals,
Hunt processes,
Equilibrium potentialsNature: Original Retrieve article from Numdam
XIV: 45, 437-474, LNM 784 (1980)
TAKSAR, Michael I.
Regenerative sets on real line (
Markov processes,
Renewal theory)
From the introduction: A number of papers are devoted to studying regenerative sets on a positive half-line... our objective is to construct translation invariant sets of this type on the entire real line. Besides we start from a weaker definition of regenerativity
Comment: This important paper, if written in recent years, would have merged into the theory of Kuznetsov measures
Keywords: Regenerative setsNature: Original Retrieve article from Numdam
XIV: 46, 475-488, LNM 784 (1980)
WEBER, Michel
Sur un théorème de Maruyama (
Gaussian processes,
Ergodic theory)
Given a stationary centered Gaussian process $X$ with spectral measure $\mu$, a new proof is given of the fact that if $\mu$ is continuous, the flow of $X$ is weakly mixing
Keywords: Stationary processesNature: Original Retrieve article from Numdam
XIV: 47, 489-495, LNM 784 (1980)
CAIROLI, Renzo
Intégrale stochastique curviligne le long d'une courbe rectifiable (
Several parameter processes)
The problem is to define stochastic integrals $\int_{\partial A} \phi\,\partial_1W$ where $W$ is the Brownian sheet, $\phi$ is a suitable process, and $A$ a suitable domain of the plane with rectifiable boundary
Keywords: Stochastic integrals,
Brownian sheetNature: Original Retrieve article from Numdam
XIV: 48, 496-499, LNM 784 (1980)
COCOZZA-THIVENT, Christiane;
YOR, Marc
Démonstration d'un théorème de F. Knight à l'aide de martingales exponentielles (
Martingale theory)
This is a new proof of Knight's theorem that (roughly) finitely many orthogonal continuous local martingales, when separately time-changed into Brownian motions, become independent. A similar theorem for the Poisson case is proved in the same way
Comment: See
518 for an earlier proof
Keywords: Changes of timeNature: Original Retrieve article from Numdam
XIV: 49, 500-546, LNM 784 (1980)
LENGLART, Érik
Tribus de Meyer et théorie des processus (
General theory of processes,
Stochastic calculus)
The subject of this paper is the study of the $\sigma$-field on $
R_+\times\Omega$ generated by a family of cadlag processes including the deterministic ones, and stable under stopping at non-random times. Of course the optional and previsible $\sigma$-fields are Meyer $\sigma$-fields in this very general sense. It is a matter of wonder to see how far one can go with such simple hypotheses, which were suggested by Dellacherie
705Comment: This beautiful paper was generally ignored. If a suggestive name had been used instead of the terminology ``Meyer $\sigma$-field'', its fate might have been different. See
1524 for an interesting application. The work of Fourati (partly unpublished) follows along the same lines, but including time reversal: see
2119Keywords: Projection theorems,
Section theoremsNature: Original Retrieve article from Numdam
XV: 01, 1-5, LNM 850 (1981)
FERNIQUE, Xavier
Sur les lois de certaines intégrales associées à des mouvements browniens (
Brownian motion)
Let $(Z_n)$ be a sequence of independent standard Brownian motions. Define by induction a sequence of processes $U_k$ by $U_0=Z_0$, $U_k(t)=\int_0^tU_{k-1}(s)dZ_k(s)$. Let $g_k(x)$ be the density of the random variable $U_k(1)$. Then the decrease at infinity of $g_k(x)$ is of the order $\exp(-C|x|^{\alpha})$ with $\alpha=2/(k+1)$ (slightly incorrect statement, see the paper for details)
Keywords: Iterated stochastic integralsNature: Original Retrieve article from Numdam
XV: 03, 11-37, LNM 850 (1981)
LEDOUX, Michel
La loi du logarithme itéré bornée dans les espaces de Banach (
Banach space valued random variables)
To be completed
Keywords: Law of the iterated logarithmNature: Original Retrieve article from Numdam
XV: 04, 38-43, LNM 850 (1981)
NOBELIS, Photis
Fonctions aléatoires lipschitziennes (
Regularity of random processes)
A sufficient condition is given so that almost all sample functions of a random process defined on $[0,1]^
N$ satisfy a Lipschitz condition (involving a general modulus of continuity). The method is that of majorizing measures. A condition due to Ibragimov is extended
Keywords: Majorizing measuresNature: Original Retrieve article from Numdam
XV: 05, 44-102, LNM 850 (1981)
MEYER, Paul-André
Géométrie stochastique sans larmes (
Stochastic differential geometry)
Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is
continuous semimartingales in manifolds, following L.~Schwartz (LN
780, 1980), but with additional features: an indication of J.M.~Bismut hinting to a definition of continuous
martingales in a manifold, and the author's own interest on the forgotten intrinsic definition of the second differential $d^2f$ of a function. All this fits together into a geometric approach to semimartingales, and a probabilistic approach to such geometric topics as torsion-free connexions
Comment: A short introduction by the same author can be found in
Stochastic Integrals, Springer LNM 851. The same ideas are expanded and presented in the supplement to Volume XVI and the book by Émery,
Stochastic Calculus on Manifolds Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Transfer principle,
Stochastic differential equations,
Stochastic integrals,
Stratonovich integralsNature: Original Retrieve article from Numdam
XV: 06, 103-117, LNM 850 (1981)
MEYER, Paul-André
Flot d'une équation différentielle stochastique (
Stochastic calculus)
Malliavin showed very neatly how an (Ito) stochastic differential equation on $
R^n$ with $C^{\infty}$ coefficients, driven by Brownian motion, generates a flow of diffeomorphisms. This consists of three results: smoothness of the solution as a function of its initial point, showing that the mapping is 1--1, and showing that it is onto. The last point is the most delicate. Here the results are extended to stochastic differential equations on $
R^n$ driven by continuous semimartingales, and only partially to the case of semimartingales with jumps. The essential argument is borrowed from Kunita and Varadhan (see Kunita's talk in the Proceedings of the Durham Symposium on SDE's, LN 851)
Comment: The results on semimartingales with jumps have been proved independently by Uppman. Some dust has been swept under the rugs about the non-explosion of the solution, and the results should be considered valid only in the globally Lipschitz case. See also Uppman
1624 and Léandre
1922Keywords: Stochastic differential equations,
Flow of a s.d.e.Nature: Exposition,
Original additions Retrieve article from Numdam
XV: 07, 118-141, LNM 850 (1981)
KUNITA, Hiroshi
Some extensions of Ito's formula (
Stochastic calculus)
The standard Ito formula expresses the composition of a smooth function $f$ with a continuous semimartingale as a stochastic integral, thus implying that the composition itself is a semimartingale. The extensions of Ito formula considered here deal with more complicated composition problems. The first one concerns a composition Let $(F(t, X_t)$ where $F(t,x)$ is a continuous semimartingale depending on a parameter $x\in
R^d$ and satisfying convenient regularity assumptions, and $X_t$ is a semimartingale. Typically $F(t,x)$ will be the flow of diffeomorphisms arising from a s.d.e. with the initial point $x$ as variable. Other examples concern the parallel transport of tensors along the paths of a flow of diffeomorphisms, or the pull-back of a tensor field by the flow itself. Such formulas (developed also by Bismut) are very useful tools of stochastic differential geometry
Keywords: Stochastic differential equations,
Flow of a s.d.e.,
Change of variable formula,
Stochastic parallel transportNature: Original Retrieve article from Numdam
XV: 09, 143-150, LNM 850 (1981)
FÖLLMER, Hans
Calcul d'Ito sans probabilités (
Stochastic calculus)
It is shown that if a deterministic continuous curve has a ``quadratic variation'' in a suitable sense (which however depends explicitly on a nested sequence of time subdivisions, for example the standard dyadic one), then it satisfies a deterministic ``Ito formula'' when composed with a twice differentiable function. Thus the only place where probability really appears in the derivation of Ito's formula is in the fact that, given any sequence of subdivisions, almost every path of a semimartingale admits a quadratic variation relative to this sequence (though no path may exist which has a quadratic variation relative to all sequences)
Comment: This subject is developed by T. Lyons' work on differential equations driven by non-smooth functions (in
Rev. Math. Iberoamericana 14, 1998)
Keywords: Stochastic integrals,
Change of variable formula,
Quadratic variationNature: Original Retrieve article from Numdam
XV: 10, 151-166, LNM 850 (1981)
MEYER, Paul-André
Retour sur la théorie de Littlewood-Paley (
Applications of martingale theory,
Markov processes)
The word ``original'' may be considered misleading, since this paper is essentially a re-issue of
1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (
J. Funct. Anal., 38, 1980)
Comment: See an application to the Ornstein-Uhlenbeck semigroup
1816, see
1818 for a related topic, and the report
1908 on Cowling's extension of Stein's work. Bouleau-Lamberton
2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry
1910, see also Bakry-Émery
1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)
Keywords: Littlewood-Paley theory,
Semigroup theory,
Riesz transforms,
Brownian motion,
Inequalities,
Harmonic functions,
Singular integrals,
Carré du champNature: Original Retrieve article from Numdam
XV: 11, 167-188, LNM 850 (1981)
BOULEAU, Nicolas
Propriétés d'invariance du domaine du générateur infinitésimal étendu d'un processus de Markov (
Markov processes)
The main result of the paper of Kunita (
Nagoya Math. J.,
36, 1969) showed that the domain of the extended generator $A$ of a right Markov semigroup is an algebra if and only if the angle brackets of all martingales are absolutely continuous with respect to the measure $dt$. See also
1010. Such semigroups are called here ``semigroups of Lebesgue type''. Kunita's result is sharpened here: it is proved in particular that if some non-affine convex function $f$ operates on the domain, then the semigroup is of Lebesgue type (Kunita's result corresponds to $f(x)=x^2$) and if the second derivative of $f$ is not absolutely continuous, then the semigroup has no diffusion part (i.e., all martingales are purely discontinuous). The second part of the paper is devoted to the behaviour of the extended domain under an absolutely continuous change of probability (arising from a multiplicative functional)
Keywords: Semigroup theory,
Carré du champ,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
XV: 12, 189-190, LNM 850 (1981)
BARLOW, Martin T.
On Brownian local time (
Brownian motion)
Let $(L^a_t)$ be the standard (jointly continuous) version of Brownian local times. Perkins has shown that for fixed $t$ $a\mapsto L^a_t$ is a semimartingale relative to the excusrion fields. An example is given of a stopping time $T$ for which $a\mapsto L^a_T$ is not a semimartingale
Keywords: Local timesNature: Original Retrieve article from Numdam
XV: 13, 191-205, LNM 850 (1981)
MAISONNEUVE, Bernard
On Lévy's downcrossing theorem and various extensions (
Excursion theory)
Lévy's downcrossing theorem describes the local time $L_t$ of Brownian motion at $0$ as the limit of $\epsilon D_t(\epsilon)$, where $D_t$ denotes the number of downcrossings of the interval $(0,\epsilon)$ up to time $t$. To give a simple proof of this result from excursion theory, easy to generalize, the paper uses the weaker definition of regenerative systems described in
1137. A gap in the related author's paper
Zeit. für W-Theorie, 52, 1980 is repaired at the end of the paper
Keywords: Excursions,
Lévy's downcrossing theorem,
Local times,
Regenerative systemsNature: Original Retrieve article from Numdam
XV: 15, 210-226, LNM 850 (1981)
JEULIN, Thierry;
YOR, Marc
Sur les distributions de certaines fonctionnelles du mouvement brownien (
Brownian motion)
This paper gives new proofs and extensions of results due to Knight, concerning occupation times by the process $(S_t,B_t)$ up to time $T_a$, where $(B_t)$ is Brownian motion, $T_a$ the hitting time of $a$, and $(S_t)$ is $\sup_{s\le t} B_s$. The method uses enlargement of filtrations, and martingales similar to those of
1306. Theorem 3.7 is a decomposition of Brownian paths akin to Williams' decomposition
Comment: See also
1516Keywords: Explicit laws,
Occupation times,
Enlargement of filtrations,
Williams decompositionNature: Original Retrieve article from Numdam
XV: 16, 227-250, LNM 850 (1981)
ROGERS, L.C.G.
Williams' characterization of the Brownian excursion law: proof and applications (
Brownian motion)
In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams
Diffusions, Markov Processes and Martingales, Wiley 1979, II.67), and it collects a number of applications, including the Azéma-Yor approach to Skorohod's imbedding theorem (
1306)
Keywords: Excursions,
Explicit laws,
Bessel processes,
Skorohod imbeddingNature: Original Retrieve article from Numdam
XV: 17, 251-258, LNM 850 (1981)
PITMAN, James W.
A note on $L_2$ maximal inequalities (
Martingale theory)
This paper contains a $L^2$ inequality between two processes $(X_n,M_n)$ under assumptions which (if $X$ is a martingale) apply to $M_n=\sup_{m\le n} |X_m|$, and to other interesting cases as well. In particular, Doob's inequality is valid for the larger process $\sup_{m\le n} X_m^+ +\sup_{m\le n} X_m^-$
Keywords: Maximal inequality,
Doob's inequalityNature: Original Retrieve article from Numdam
XV: 18, 259-277, LNM 850 (1981)
BRU, Bernard;
HEINICH, Henri;
LOOTGIETER, Jean-Claude
Autour de la dualité $(H^1,BMO)$ (
Martingale theory)
This is a sequel to
1330. Given two martingales $(X,Y)$ in $H^1$ and $BMO$, it is investigated whether their duality functional can be safely estimated as $E[X_{\infty}Y_{\infty}]$. The simple result is that if $X_{\infty}Y_{\infty}$ belongs to $L^1$, or merely is bounded upwards by an element of $L^1$, then the answer is positive. The second (and longer) part of the paper searches for subspaces of $H^1$ and $BMO$ such that the property would hold between their elements, and here the results are fragmentary (a question of
1330 is answered). An appendix discusses a result of Talagrand
Keywords: $BMO$,
$H^1$ space,
Hardy spacesNature: Original Retrieve article from Numdam
XV: 19, 278-284, LNM 850 (1981)
ÉMERY, Michel
Le théorème de Garnett-Jones, d'après Varopoulos (
Martingale theory)
Let $M$ be a martingale belonging to $BMO$. The John-Nirenberg theorem implies that, for some constant $0<\lambda<\infty$, the conditional expectations $E[\exp( {1\over\lambda}(M_{\infty} -M_{T_-}))\, |\,{\cal F}_T]$ belongs to $L^{\infty}$ for all stopping times $T$, with a norm independent of $T$. The Garnett-Jones theorem (proved by Varopoulos in the probabilistic set-up) asserts that the smallest such $\lambda$ is ``equivalent'' to the $BMO$ distance of $M$ to the subspace $L^\infty$. One half of the equivalence is general, while the other half requires all martingales of the filtration to be continuous. The examples given in the second part show that this hypothesis is essential
Keywords: $BMO$Nature: Exposition,
Original additions Retrieve article from Numdam
XV: 20, 285-289, LNM 850 (1981)
CHOU, Ching Sung
Une inégalité de martingales avec poids (
Martingale theory)
Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in
1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see
1326)
Keywords: Weighted norm inequalities,
Burkholder inequalities,
Moderate convex functionsNature: Original Retrieve article from Numdam
XV: 21, 290-306, LNM 850 (1981)
CHACON, Rafael V.;
LE JAN, Yves;
WALSH, John B.
Spatial trajectories (
Markov processes,
General theory of processes)
It is well known that Markov processes with the same excessive functions are the same up to a strictly increasing continuous time-change. It is therefore natural to study spatial trajectories, i.e., trajectories up to a strictly increasing continuous time changes, and in particular to provide the space of all spatial trajectories with a reasonable $\sigma$-field so that it may carry measures. It is shown here that the space of right-continuous spatial trajectories with left-hand limits is a Blackwell space. The class of intrinsic stopping times defined on this space is also investigated
Comment: See Chacon-Jamison,
Israel J. of M.,
33, 1979
Keywords: Spatial trajectoriesNature: Original Retrieve article from Numdam
XV: 22, 307-310, LNM 850 (1981)
LE JAN, Yves
Tribus markoviennes et prédiction (
Markov processes,
General theory of processes)
The problem discussed here is whether a given filtration is generated by a Ray process. The answer is positive under very general conditions. Knight's prediction theory (
1007) is used
Keywords: Prediction theoryNature: Original Retrieve article from Numdam
XV: 23, 311-319, LNM 850 (1981)
ALDOUS, David J.;
BARLOW, Martin T.
On countable dense random sets (
General theory of processes,
Point processes)
This paper is devoted to random sets $B$ which are countable, optional (i.e., can be represented as the union of countably many graphs of stopping times $T_n$) and dense. The main result is that whenever the increasing processes $I_{t\ge T_n}$ have absolutely continuous compensators (in which case the same property holds for any stopping time $T$ whose graph is contained in $B$), then the random set $B$ can be represented as the union of all the points of countably many independent standard Poisson processes (intuitively, a Poisson measure whose rate is $+\infty$ times Lebesgue measure). This may require, however, an innocuous enlargement of filtration. Another characterization of such random sets is roughly that they do not intersect previsible sets of zero Lebesgue measure. Note also an interesting example of a set optional w.r.t. two filtrations, but not w.r.t. their intersection
Keywords: Poisson point processesNature: Original Retrieve article from Numdam
XV: 24, 320-346, LNM 850 (1981)
DELLACHERIE, Claude;
LENGLART, Érik
Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des martingales (
General theory of processes)
The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in
1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)
Keywords: General filtrations,
Strong supermartingales,
Snell's envelope,
Section theoremsNature: Original Retrieve article from Numdam
XV: 25, 347-350, LNM 850 (1981)
MAISONNEUVE, Bernard
Surmartingales-mesures (
Martingale theory)
Consider a discrete filtration $({\cal F}_n)$ and let ${\cal A}$ be the algebra, $\cup_n {\cal F}_n$, generating a $\sigma$-algebra ${\cal F}_\infty$. A positive supermartingale $(X_n)$ is called a supermartingale measure if the set function $A\mapsto\lim_n\int_A X_n\,dP$ on $A$ is $\sigma$-additive, and thus can be extended to a measure $\mu$. Then the Lebesgue decomposition of this measure is described (theorem 1). More generally, the Lebesgue decomposition of any measure $\mu$ on ${\cal F}_\infty$ is described. This is meant to complete theorem III.1.5 in Neveu,
Martingales à temps discret Comment: The author points out at the end that theorem 2 had been already proved by Horowitz (
Zeit. für W-theorie, 1978) in continuous time. This topic is now called Kunita decomposition, see
1005 and the corresponding references
Keywords: Supermartingales,
Kunita decompositionNature: Original Retrieve article from Numdam
XV: 26, 351-370, LNM 850 (1981)
DELLACHERIE, Claude
Mesurabilité des débuts et théorème de section~: le lot à la portée de toutes les bourses (
General theory of processes)
One of the main topics in these seminars has been the application to stochastic processes of results from descriptive set theory and capacity theory, at different levels. Since these results are considered difficult, many attempts have been made to shorten and simplify the exposition. A noteworthy one was
511, in which Dellacherie introduced ``rabotages'' (
306) to develop the theory without analytic sets; see also
1246,
1255. The main feature of this paper is a new interpretation of rabotages as a two-persons game, ascribed to Telgarsky though no reference is given, leading to a pleasant exposition of the whole theory and its main applications
Keywords: Section theorems,
Capacities,
Sierpinski's ``rabotages''Nature: Original Retrieve article from Numdam
XV: 27, 371-387, LNM 850 (1981)
DELLACHERIE, Claude
Sur les noyaux $\sigma$-finis (
Measure theory)
This paper is an improvement of
1235. Assume $(X,{\cal X})$ and $(Y,{\cal Y})$ are measurable spaces and $m(x,A)$ is a kernel, i.e., is measurable in $x\in X$ for $A\in{\cal Y}$, and is a $\sigma$-finite measure in $A$ for $x\in X$. Then the problem is to represent the measures $m(x,dy)$ as $g(x,y)\,N(x,dy)$ where $g$ is a jointly measurable function and $N$ is a Markov kernel---possibly enlarging the $\sigma$-field ${\cal X}$ to include analytic sets. The crucial hypothesis (called
measurability of $m$) is the following: for every auxiliary space $(Z, {\cal Z})$, the mapping $(x,z)\mapsto m_x\otimes \epsilon_z$ is again a kernel (in fact, the auxiliary space $
R$ is all one needs). The case of ``basic'' kernels, considered in
1235, is thoroughly discussed
Keywords: Kernels,
Radon-Nikodym theoremNature: Original Retrieve article from Numdam
XV: 29, 399-412, LNM 850 (1981)
YOEURP, Chantha
Sur la dérivation stochastique au sens de Davis (
Stochastic calculus,
Brownian motion)
The problem is that of estimating $H_t$ from a knowledge of the process $\int H_sdX_s$, where $X_t$ is a continuous (semi)martingale, as a limit of ratios of the form $\int_t^{t+\alpha} H_sdX_s/(X_{t+\alpha}-X_t)$, or replacing $t+\alpha$ by suitable stopping times
Comment: The problem was suggested and partially solved by Mark H.A. Davis (
Teor. Ver. Prim.,
20, 1975, 887--892). See also
1428Keywords: Stochastic integralsNature: Original Retrieve article from Numdam
XV: 30, 413-489, LNM 850 (1981)
SCHWARTZ, Laurent
Les semi-martingales formelles (
Stochastic calculus,
General theory of processes)
This is a natural development of the 1--1correspondence between semimartingales and $\sigma$-additive $L^0$-valued vector measures on the previsible $\sigma$-field, which satisfy a suitable boundedness property. What if boundedness is replaced by a $\sigma$-finiteness property? It turns out that these measures can be represented as formal stochastic integrals $H{\cdot} X$ where $X$ is a standard semimartingale, and $H$ is a (finitely valued, but possibly non-integrable) previsible process. The basic definition is quite elementary: $H{\cdot}X$ is an equivalence class of pairs $(H,X)$, where two pairs $(H,X)$ and $(K,Y)$ belong to the same class iff for some (hence for all) bounded previsible process $U>0$ such that $LH$ and $LK$ are bounded, the (usual) stochastic integrals $(UH){\cdot}X$ and $(UK){\cdot}Y$ are equal. (One may take for instance $U=1/(1{+}|H|{+}|K|)$.)\par As a consequence, the author gives an elegant and pedagogical characterization of the space $L(X)$ of all previsible processes integrable with respect to $X$ (introduced by Jacod,
1126; see also
1415,
1417 and
1424). This works just as well in the case when $X$ is vector-valued, and gives a new definition of vector stochastic integrals (see Galtchouk,
Proc. School-Seminar Vilnius, 1975, and Jacod
1419). \par Some topological considerations (that can be skipped if the reader is not interested in convergences of processes) are delicate to follow, specially since the theory of unbounded vector measures (in non-locally convex spaces!) requires much care and is difficult to locate in the literature
Keywords: Semimartingales,
Formal semimartingales,
Stochastic integralsNature: Original Retrieve article from Numdam
XV: 31, 490-492, LNM 850 (1981)
STRICKER, Christophe
Sur deux questions posées par Schwartz (
Stochastic calculus)
Schwartz studied semimartingales in random open sets, and raised two questions: Given a semimartingale $X$ and a random open set $A$, 1) Assume $X$ is increasing in every subinterval of $A$; then is $X$ equal on $A$ to an increasing adapted process on the whole line? 2) Same statement with ``increasing'' replaced by ``continuous''. Schwartz could prove statement 1) assuming $X$ was continuous. It is proved here that 1) is false if $X$ is only cadlag, and that 2) is false in general, though it is true if $A$ is previsible, or only accessible
Keywords: Random sets,
Semimartingales in a random open setNature: Original Retrieve article from Numdam
XV: 32, 493-498, LNM 850 (1981)
STRICKER, Christophe
Quasi-martingales et variations (
Martingale theory)
This paper contains remarks on quasimartingales, the most useful of which being perhaps the fact that, for a right-continuous process, the stochastic variation is the same with respect to the filtrations $({\cal F}_{t})$ and $({\cal F}_{t-})$
Keywords: QuasimartingalesNature: Original Retrieve article from Numdam
XV: 33, 499-522, LNM 850 (1981)
STRICKER, Christophe
Quelques remarques sur la topologie des semimartingales. Applications aux intégrales stochastiques (
Stochastic calculus)
This paper contains a number of useful technical results on the topology of semimartingales (see
1324), some of which were previously known with more complicated proofs. In particular, it is shown how to improve the convergence of sequences of semimartingales by a convenient change of probability. The topology of semimartingales is used to handle elegantly the stochastic integration of previsible processes which are not locally bounded (see
1415). Finally, boundedness of a set of semimartingales is shown to be equivalent to the boundedness (in an elementary sense) of a set of increasing processes controlling them in the sense of Métivier-Pellaumail (see
1412,
1413,
1414)
Keywords: Semimartingales,
Stochastic integrals,
Spaces of semimartingales,
Métivier-Pellaumail inequalityNature: Original Retrieve article from Numdam
XV: 34, 523-525, LNM 850 (1981)
STRICKER, Christophe
Sur la caractérisation des semi-martingales (
General theory of processes,
Stochastic calculus)
This is a sequel to the preceding paper
1533, giving a simple proof that any semimartingale may be brought into any class ${\cal S}^p$ by a convenient change of probability
Keywords: Semimartingales,
Spaces of semimartingalesNature: Original Retrieve article from Numdam
XV: 35, 526-528, LNM 850 (1981)
YOR, Marc
Sur certains commutateurs d'une filtration (
General theory of processes)
Let $({\cal F}_t)$ be a filtration satisfying the usual conditions and ${\cal G}$ be a $\sigma$-field. Then the conditional expectation $E[.|{\cal G}]$ commutes with $E[.|{\cal F}_T]$ for all stopping times $T$ if and only if for some stopping time $S$ ${\cal G}$ lies between ${\cal F}_{S-}]$ and ${\cal F}_S]$
Keywords: Conditional expectationsNature: Original Retrieve article from Numdam
XV: 36, 529-546, LNM 850 (1981)
JACOD, Jean;
MÉMIN, Jean
Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité (
Measure theory)
For simplicity we consider only real valued r.v.'s, but it is essential that the paper considers general Polish spaces instead of $
R$. Let us define a fuzzy r.v. $X$ on $(\Omega, {\cal F},P)$ as a probability measure on $\Omega\times
R$ whose projection on $\Omega$ is $P$. In particular, a standard r.v. $X$ defines such a measure as the image of $P$ under the map $\omega\mapsto (\omega,X(\omega))$. The space of fuzzy r.v.'s is provided with a weak topology, associated with the bounded functions $f(\omega,x)$ which are continuous in $x$ for every $\omega$, or equivalently with the functions $I_A(\omega)\,f(x)$ with $f$ bounded continuous. The main topic of this paper is the study of this topology
Comment: From this description, it is clear that this paper extends to general Polish spaces the topology of Baxter-Chacon (forgetting about the filtration), for which see
1228Keywords: Fuzzy random variables,
Convergence in lawNature: Original Retrieve article from Numdam
XV: 37, 547-560, LNM 850 (1981)
JACOD, Jean
Convergence en loi de semimartingales et variation quadratique (
General theory of processes,
Stochastic calculus)
The convergence in law of cadlag processes to a cadlag process being understood in the sense of Skorohod, the problem is to find sufficient conditions under which, given semimartingales $X^n$ and $X$ such that $X^n\rightarrow X$ in law, one may deduce that $[X^n,X^n]$ converges in law to $[X,X]$. This is achieved assuming a uniform bound on the expectations of the supremum of the jumps. A version of the theorem applied to processes which are not semimartingales, but are equal to semimartingales on large sets
Keywords: Semimartingales,
Skorohod topology,
Convergence in lawNature: Original Retrieve article from Numdam
XV: 38, 561-586, LNM 850 (1981)
PELLAUMAIL, Jean
Solutions faibles et semi-martingales (
Stochastic calculus,
General theory of processes)
From the author's summary: ``we consider a stochastic differential equation $dX=a(X)\,dZ$ where $Z$ is a semimartingale and $a$ is a previsible functional which is continuous for the uniform norm. We prove the existence of a weak solution for such an equation''. The important point is the definition of a weak solution: it turns out to be a ``fuzzy process'' in the sense of
1536, i.e., a fuzzy r.v. taking values in the Polish space of cadlag sample functions
Keywords: Stochastic differential equations,
Weak solutions,
Fuzzy random variablesNature: Original Retrieve article from Numdam
XV: 39, 587-589, LNM 850 (1981)
ÉMERY, Michel
Non-confluence des solutions d'une équation stochastique lipschitzienne (
Stochastic calculus)
This paper proves that the solutions of a stochastic differential equation $dX_t=f(., t,X_t)\,dM_t$ driven by a continuous semimartingale $M$, where $f(\omega,t,x)$ is as usual previsible in $\omega$ and Lipschitz in $x$, are non-confluent, i.e., the solutions starting at different points never meet
Comment: See also
1506,
1507 (for less general s.d.e.'s), and
1624Keywords: Stochastic differential equations,
Flow of a s.d.e.Nature: Original Retrieve article from Numdam
XV: 40, 590-603, LNM 850 (1981)
STROOCK, Daniel W.;
YOR, Marc
Some remarkable martingales (
Martingale theory)
This is a sequel to a well-known paper by the authors (
Ann. ENS, 13, 1980) on the subject of pure martingales. A continuous martingale $(M_t)$ with $<M,M>_{\infty}=\infty$ is pure if the time change which reduces it to a Brownian motion $(B_t)$ entails no loss of information, i.e., if $M$ is measurable w.r.t. the $\sigma$-field generated by $B$. The first part shows the purity of certain stochastic integrals. Among the striking examples considered, the stochastic integrals $\int_0^t B^n_sdB_s$ are extremal for every integer $n$, pure for $n$ odd, but nothing is known for $n$ even. A beautiful result unrelated to purity is the following: complex Brownian motion $Z_t$ starting at $z_0$ and its (Lévy) area integral generate the same filtration if and only if $z_0\neq0$
Keywords: Pure martingales,
Previsible representationNature: Original Retrieve article from Numdam
XV: 41, 604-617, LNM 850 (1981)
LÉPINGLE, Dominique;
MEYER, Paul-André;
YOR, Marc
Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (
General theory of processes,
Martingale theory)
This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in
1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case
Keywords: Poisson processes,
Pure martingales,
Previsible representation,
JumpsNature: Original Retrieve article from Numdam
XV: 42, 618-626, LNM 850 (1981)
ITMI, Mhamed
Processus ponctuels marqués stochastiques. Représentation des martingales et filtration naturelle quasicontinue à gauche (
General theory of processes)
This paper contains a study of the filtration generated by a point process (multivariate: it takes values in a Polish space), and in particular of its quasi-left continuity, and previsible representation
Keywords: Point processes,
Previsible representationNature: Original Retrieve article from Numdam
XV: 43, 627-631, LNM 850 (1981)
WANG, Jia-Gang
Some remarks on processes with independent increments (
Independent increments)
This paper contains results on non-homogeneous processes with independent increments, without fixed discontinuities, which belong to the folklore of the subject but are hard to locate in the literature. The first one is that their natural filtration, merely augmented by all sets of measure $0$, is automatically right-continuous and quasi-left-continuous. The second one concerns those processes which are multivariate point processes, i.e., have only finitely many jumps in finite intervals and are constant between jumps. It is shown how to characterize the independent increments property into a property of the process of jumps conditioned by the process of jump times. Finally, a remark is done to the order that several results extend automatically to random measures with independent increments, for which see also
1544Keywords: Poisson processes,
Lévy measuresNature: Original Retrieve article from Numdam
XV: 45, 643-668, LNM 850 (1981)
AUERHAN, J.;
LÉPINGLE, Dominique
Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (
General theory of processes,
Brownian motion,
Martingale theory)
This is a sequel to
1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in
1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given
Comment: Further results are given by Malric
Ann. Inst. H. Poincaré 26 (1990)
Nature: Original Retrieve article from Numdam
XV: 47, 671-672, LNM 850 (1981)
BAKRY, Dominique
Une remarque sur les semi-martingales à deux indices (
Several parameter processes)
Let $({\cal F}^1_s)$ and $({\cal F}^2_t)$ be two filtrations whose conditional expectations commute. Let $(A_t)$ be a bounded increasing process adapted to $({\cal F}^2_t)$. It had been proved under stringent absolute continuity conditions on $A$ that the process $X_{st}=E[A_t\,|\,{\cal F}^1_s]$ was a semimartingale (a stochastic integrator). A counterexample is given here to show that this is not true in general
Keywords: Two-parameter semimartingalesNature: Original Retrieve article from Numdam
XV: 48, 673-688, LNM 850 (1981)
MAZZIOTTO, Gérald;
SZPIRGLAS, Jacques
Un exemple de processus à deux indices sans l'hypothèse F4 (
Several parameter processes)
A natural two-parameter filtration is associated with a random point in the positive quadrant $
R^2_+$. Though it does not satisfy in general the Cairoli-Walsh commutation property called F4, it is possible to develop for this filtration a reasonable theory of optional and previsible processes, projection theorems, etc
Keywords: Two-parameter processesNature: Original Retrieve article from Numdam
XVI: 06, 95-132, LNM 920 (1982)
MEYER, Paul-André
Note sur les processus d'Ornstein-Uhlenbeck (
Malliavin's calculus)
With every Gaussian measure $\mu$ one can associate an Ornstein-Uhlenbeck semigroup, for which $\mu$ is a reversible invariant measure. When $\mu$ is Wiener's measure on ${\cal C}(
R)$, this semigroup is a fundamental tool in Malliavin's own approach to the ``Malliavin calculus''. See for instance Stroock's exposition of it in
Math. Systems Theory, 13, 1981. With this semigroup one can associate its generator $L$ which plays the role of the classical Laplacian, and the positive bilinear functional $\Gamma(f,g)= L(fg)-fLg-gLf$---leaving aside domain problems for simplicity---sometimes called ``carré du champ'', which plays the role of the squared classical gradient. As in classical analysis, one can define it as $\sum_i \nabla_i f\nabla i g$, the derivatives being relative to an orthonormal basis of the Cameron-Martin space. We may define Sobolev-like spaces of order one in two ways: either by the fact that $Cf$ belongs to $L^p$, where $C=-\sqrt{-L}$ is the ``Cauchy generator'', or by the fact that $\sqrt{\Gamma(f,f)}$ belongs to $L^p$. A result which greatly simplifies the analytical part of the ``Malliavin calculus'' is the fact that both definitions are equivalent. This is the main topic of the paper, and its proof uses the Littlewood-Paley-Stein theory for semigroups as presented in
1010,
1510Comment: An important problem is the extension to higher order Sobolev-like spaces. For instance, we could define the Sobolev space of order 2 either by the fact that $C^2f=-Lf$ belongs to $L^p$, and on the other hand define $\Gamma_2(f,g)=\sum_{ij} \nabla_i\nabla_j f \nabla_i\nabla_j g$ (derivatives of order 2) and ask that $\sqrt{\Gamma_2(f,f)}\in L^p$. For the equivalence of these two definitions and general higher order ones, see
1816, which anyhow contains many improvements over
1606. Also, proofs of these results have been given which do not involve Littlewood-Paley methods. For instance, Pisier has a proof which only uses the boundedness in $L^p$ of classical Riesz transforms.\par Another trend of research has been the correct definition of ``higher gradients'' within semigroup theory (the preceding definition of $\Gamma_2(f,g)$ makes use of the Gaussian structure). Bakry investigated the fundamental role of ``true'' $\Gamma_2$, the bilinear form $\Gamma_2(f,g)=L\Gamma(f,g)-\Gamma(Lf,g)-\Gamma(Lf,g)$, which is positive in the case of the Ornstein-Uhlenbeck semigroup but is not always so. See
1909,
1910,
1912Keywords: Ornstein-Uhlenbeck process,
Gaussian measures,
Littlewood-Paley theory,
Hypercontractivity,
Hermite polynomials,
Riesz transforms,
Test functionsNature: Exposition,
Original additions Retrieve article from Numdam
XVI: 08, 134-137, LNM 920 (1982)
BAKRY, Dominique
Remarques sur le processus d'Ornstein-Uhlenbeck en dimension infinie (
Malliavin's calculus,
Several parameter processes)
A process taking values in a space of sample paths can be considered as a two parameter process. Considering in this way the Ornstein-Uhlenbeck process (
1606) raises a few natural questions, like the commutation of conditional expectations relative to the two filtrations---which is shown to hold true
Keywords: Ornstein-Uhlenbeck processNature: Original Retrieve article from Numdam
XVI: 09, 138-150, LNM 920 (1982)
BAKRY, Dominique;
MEYER, Paul-André
Sur les inégalités de Sobolev logarithmiques (two parts) (
Applications of martingale theory)
These two papers are variations on a paper of G.F. Feissner (
Trans. Amer Math. Soc.,
210, 1965). Let $\mu$ be a Gaussian measure, $P_t$ be the corresponding Ornstein-Uhlenbeck semigroup. Nelson's hypercontractivity theorem states (roughly) that $P_t$ is bounded from $L^p(\mu)$ to some $L^q(\mu)$ with $q\ge p$. In another celebrated paper, Gross showed this to be equivalent to a logarithmic Sobolev inequality, meaning that if a function $f$ is in $L^2$ as well as $Af$, where $A$ is the Ornstein-Uhlenbeck generator, then $f$ belongs to the Orlicz space $L^2Log_+L$. The starting point of Feissner was to translate this again as a result on the ``Riesz potentials'' of the semi-group (defined whenever $f\in L^2$ has integral $0$) $$R^{\alpha}={1\over \Gamma(\alpha)}\int_0^\infty t^{\alpha-1}P_t\,dt\;.$$ Note that $R^{\alpha}R^{\beta}=R^{\alpha+\beta}$. Then the theorem of Gross implies that $R^{1/2}$ is bounded from $L^2$ to $L^2Log_+L$. This suggests the following question: which are in general the smoothing properties of $R^\alpha$? (Feissner in fact considers a slightly different family of potentials).\par The complete result then is the following : for $\alpha$ complex, with real part $\ge0$, $R^\alpha$ is bounded from $L^pLog^r_+L$ to $L^pLog^{r+p\alpha}_+L$. The method uses complex interpolation between two cases: a generalization to Orlicz spaces of a result of Stein, when $\alpha$ is purely imaginary, and the case already known where $\alpha$ has real part $1/2$. The first of these two results, proved by martingale theory, is of a quite general nature
Keywords: Logarithmic Sobolev inequalities,
Hypercontractivity,
Gaussian measures,
Riesz potentialsNature: Original Retrieve article from Numdam
XVI: 10, 151-152, LNM 920 (1982)
MEYER, Paul-André
Sur une inégalité de Stein (
Applications of martingale theory)
In his book
Topics in harmonic analysis related to the Littlewood-Paley theory (1970) Stein uses interpolation between two results, one of which is a discrete martingale inequality deduced from the Burkholder inequalities, whose precise statement we omit. This note states and proves directly the continuous time analogue of this inequality---a mere exercise in translation
Keywords: Littlewood-Paley theory,
Martingale inequalitiesNature: Exposition,
Original additions Retrieve article from Numdam
XVI: 15, 209-211, LNM 920 (1982)
BARLOW, Martin T.
$L(B_t,t)$ is not a semimartingale (
Brownian motion)
The following question had been open for some time: given a jointly continuous version $L(a,t)$ of the local times of Brownian motion, is $Y_t=L(B_t,t)$ a semimartingale? It is proved here that $Y$ fails to be Hölder continuous of order 1/4, and therefore cannot be a semimartingale
Keywords: Local times,
SemimartingalesNature: Original Retrieve article from Numdam
XVI: 16, 212-212, LNM 920 (1982)
WALSH, John B.
A non-reversible semi-martingale (
Stochastic calculus)
A simple example is given of a continuous semimartingale (a Brownian motion which stops at time $1$ and starts moving again at time $T>1$, $T$ encoding all the information up to time $1$) whose reversed process is not a semimartingale
Keywords: Semimartingales,
Time reversalNature: Original Retrieve article from Numdam
XVI: 17, 213-218, LNM 920 (1982)
FALKNER, Neil;
STRICKER, Christophe;
YOR, Marc
Temps d'arrêt riches et applications (
General theory of processes)
This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see
1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper
1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$
Keywords: Stopping times,
Local times,
Semimartingales,
Previsible processesNature: Original Retrieve article from Numdam
XVI: 18, 219-220, LNM 920 (1982)
STRICKER, Christophe
Les intervalles de constance de $\langle X,X\rangle$ (
Martingale theory,
Stochastic calculus)
For a continuous (local) martingale $X$, the constancy intervals of $X$ and $<X,X>$ are exactly the same. What about general local martingales? It is proved that $X$ is constant on the constancy intervals of $<X,X>$, and the converse holds if $X$ has the previsible representation property
Keywords: Quadratic variation,
Previsible representationNature: Original Retrieve article from Numdam
XVI: 19, 221-233, LNM 920 (1982)
YOR, Marc
Application de la relation de domination à certains renforcements des inégalités de martingales (
Martingale theory)
The domination relation (Lenglart 1977) between a positive, right-continuous process $X$ and a previsible increasing process $A$ holds whenever $E[X_T]\le E[A_T]$ at stopping times. It plays an important role in the paper
1404 of Lenglart-Lepingle-Pratelli on martingale inequalities. Here it is shown to imply a general inequality involving $X^\ast_{\infty}$ and $1/A_{\infty}$, from which follow a number of inequalities for a continuous local martingale $M$. Among them, estimates on the ratios of the three quantities $M^\ast_{\infty}$, $<M>_{\infty}$, $\sup_{a,t} L^a_t$. One can recover also the stronger version of Doob's inequality, proved by Pitman
1517Comment: See an earlier paper of the author on this subject,
Stochastics, 3, 1979. The author mentions that part of the results were discovered slightly earlier by R.~Gundy
Keywords: Martingale inequalities,
Domination inequalitiesNature: Original Retrieve article from Numdam
XVI: 20, 234-237, LNM 920 (1982)
YOEURP, Chantha
Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (
Brownian motion,
Stochastic calculus)
A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$
Comment: See
1023,
1321Keywords: Multiplicative decomposition,
Change of variable formula,
Local timesNature: Original Retrieve article from Numdam
XVI: 21, 238-247, LNM 920 (1982)
YOR, Marc
Sur la transformée de Hilbert des temps locaux browniens et une extension de la formule d'Itô (
Brownian motion)
This paper is about the application to the function $(x-a)\log|x-a|-(x-a)$ (whose second derivative is $1/x-a$) of the Ito-Tanaka formula; the last term then involves a formal Hilbert transform $\tilde L^a_t$ of the local time process $L^a_t$. Such processes had been defined by Ito and McKean, and studied by Yamada as examples of Fukushima's ``additive functionals of zero energy''. Here it is proved, as a consequence of a general theorem, that this process has a jointly continuous version---more precisely, Hölder continuous of all orders $<1/2$ in $a$ and in $t$
Comment: For a modern version with references see Yor,
Some Aspects of Brownian Motion II, Birkhäuser 1997
Keywords: Local times,
Hilbert transform,
Ito formulaNature: Original Retrieve article from Numdam
XVI: 22, 248-256, LNM 920 (1982)
JEULIN, Thierry
Sur la convergence absolue de certaines intégrales (
General theory of processes)
This paper is devoted to the a.s. absolute convergence of certain random integrals, a classical example of which is $\int_0^t ds/|B_s|^{\alpha}$ for Brownian motion starting from $0$. The author does not claim to prove deep results, but his technique of optional increasing reordering (réarrangement) of a process should be useful in other contexts too
Comment: This paper greatly simplifies a proof in the author's
Semimartingales et Grossissement de Filtrations, LNM
833, p.44
Keywords: Enlargement of filtrationsNature: Original Retrieve article from Numdam
XVI: 23, 257-267, LNM 920 (1982)
FLIESS, Michel;
NORMAND-CYROT, Dorothée
Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T.~Chen (
Stochastic calculus)
Consider a s.d.e. in a manifold, $dX_t=\sum_i A_i(X)\,dM^i_t$ (Stratonovich differentials), driven by continuous real semimartingales $M^i_t$, and where the $A_i$ have the geometrical nature of vector fields. Such an equation has a counterpart in which the $M^i(t)$ are arbitrary deterministic piecewise smooth functions, and if this equation can be solved by some deterministic machinery, then the s.d.e. can be solved too, just by making the input random. Thus a bridge is drawn between s.d.e.'s and problems of deterministic control theory. From this point of view, the complexity of the problem reflects that of the Lie algebra generated by the vector fields $A_i$. Assuming these fields are complete (i.e., generate true one-parameter groups on the manifold) and generate a finite dimensional Lie algebra (which then is the Lie algebra of a matrix group), the problem can be linearized. If the Lie algebra is nilpotent, the solution then can be expressed explicitly as a function of a finite number of iterated integrals of the driving processes (Chen integrals), and this provides the required ``deterministic machine''. It thus appears that results like those of Yamato (
Zeit. für W-Theorie, 47, 1979) do not really belong to probability theory
Comment: See Kunita's paper
1432Keywords: Stochastic differential equations,
Lie algebras,
Chen's iterated integrals,
Campbell-Hausdorff formulaNature: Original Retrieve article from Numdam
XVI: 24, 268-284, LNM 920 (1982)
UPPMAN, Are
Sur le flot d'une équation différentielle stochastique (
Stochastic calculus)
This paper is a companion to
1506, devoted to the main results on the flow of a (Lipschitz) stochastic differential equation driven by continous semimartingales: non-confluence of solutions from different initial points, surjectivity of the mapping, smooth dependence on the initial conditions. The proofs have been greatly simplified
Keywords: Stochastic differential equations,
Flow of a s.d.e.,
InjectivityNature: Exposition,
Original additions Retrieve article from Numdam
XVI: 25, 285-297, LNM 920 (1982)
UPPMAN, Are
Un théorème de Helly pour les surmartingales fortes (
Martingale theory)
Provide the set of (optional) strong supermartingales $X$ of the class (D) with the topology of weak $L^1$--convergence of $X_T$ at each stopping time $T$. Then it is shown that any subset which belongs uniformly to the class (D) is relatively compact, also in the sequential sense of extracting convergent subsequences
Comment: This paper was suggested by a similar result of Mokobodzki for strongly supermedian functions in potential theory
Keywords: Supermartingales,
Strong supermartingalesNature: Original Retrieve article from Numdam
XVI: 26, 298-313, LNM 920 (1982)
DELLACHERIE, Claude;
LENGLART, Érik
Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (
General theory of processes)
This paper is a sequel to
1524. Let $\Theta$ be a
chronology, i.e., a family of stopping times containing $0$ and $\infty$ and closed under the operations $\land,\lor$---examples are the family of all stopping times, and that of all deterministic stopping times. The general problem discussed is that of defining an optional process $X$ on $[0,\infty]$ such that for each $T\in\Theta$ $X_T$ is a.s. equal to a given r.v. (${\cal F}_T$-measurable). While in
1525 the discussion concerned supermartingales, it is extended here to processes which satisfy a semi-continuity condition from the right
Keywords: Stopping timesNature: Original Retrieve article from Numdam
XVI: 27, 314-318, LNM 920 (1982)
LENGLART, Érik
Sur le théorème de la convergence dominée (
General theory of processes,
Stochastic calculus)
Consider previsible processes $U^n,U$ such that $U^n_T\rightarrow U_T$ in some sense at bounded previsible times $T$. The problem discussed is whether stochastic integrals $\int U^n_s dX_s$ converge (in the same sense) to $\int U_sdX_s$. Under a domination hypothesis, the answer is shown to be positive if the convergence is either weak convergence in $L^1$, or convergence in probability. The existence of the limiting process $U$ is not assumed in the paper; it is proved by a modification of an argument of Mokobodzki for which see
1110Keywords: Stopping times,
Optional processes,
Weak convergence,
Stochastic integralsNature: Original Retrieve article from Numdam
XVI: 29, 338-347, LNM 920 (1982)
YAN, Jia-An
À propos de l'intégrabilité uniforme des martingales exponentielles (
Martingale theory)
Sufficient conditions are given for the uniform integrability of the exponential ${\cal E}(M)$, where $M$ is a local martingale with jumps $\ge-1$, refining older results of Lépingle and Mémin, and of the author. They involve the Lévy measure of the martingale
Comment: In the lemma p.339 delete the assumption $0<\beta$
Keywords: Exponential martingalesNature: Original Retrieve article from Numdam
XVI: 30, 348-354, LNM 920 (1982)
HE, Sheng-Wu;
WANG, Jia-Gang
The total continuity of natural filtrations (
General theory of processes)
Total continuity of a filtration ${\cal F}$ means that ${\cal F}_T={\cal F}_{T-}$ at every stopping time $T$, not necessarily previsible. It is shown that the filtration of a Lévy process without fixed discontinuities is totally continuous if and only if the jump size is a deterministic function of the jump time. Similarly, the natural filtration of a quasi-left continuous jump process is totally continuous if and only if the size of the $n$-th jump is a deterministic function of the jump times up to the $n$-th. It is shown that under the usual (here called ``strong'') previsible representation property, quasi-left continuity of the filtration implies total continuity
Keywords: Filtrations,
Independent increments,
Previsible representation,
Total continuity,
Lévy processesNature: Original Retrieve article from Numdam
XVI-S: 57, 165-207, LNM 921 (1982)
MEYER, Paul-André
Géométrie différentielle stochastique (bis) (
Stochastic differential geometry)
A sequel to
1505. The main theme is that an ordinary differential equation has a non unique extension as a stochastic differential equation: besides the Stratonovich one, given by the ``transfer principle'', there are other possibilities: choosing among them requires some additional, connection-like, structure. The most striking application is the Dohrn-Guerra correction to the parallel transport along a semimartingale
Comment: For complements, see Émery
1658, Hakim-Dowek-Lépingle
2023, Émery's monography
Stochastic Calculus in Manifolds (Springer, 1989) and article
2428, and Arnaudon-Thalmaier
3214Keywords: Semimartingales in manifolds,
Stochastic differential equations,
Local characteristics,
Nelson's stochastic mechanics,
Transfer principleNature: Original Retrieve article from Numdam
XVI-S: 58, 208-216, LNM 921 (1982)
ÉMERY, Michel
En marge de l'exposé de Meyer : ``Géométrie différentielle stochastique'' (
Stochastic differential geometry)
Marginal remarks to Meyer
1657Keywords: Semimartingales in manifolds,
Stochastic differential equationsNature: Original Retrieve article from Numdam
XVI-S: 59, 217-236, LNM 921 (1982)
DARLING, Richard W.R.
Martingales in manifolds - Definition, examples and behaviour under maps (
Stochastic differential geometry)
Martingales in manifolds have been introduced independently by Meyer
1505 and the author (Ph.D. Thesis). This short note is a review of that thesis; here, the definition of a manifold-valued martingale is by its behaviour under convex functions
Comment: More details are given in
Bull. L.M.S. 15 (1983),
Publ R.I.M.S. Kyoto~
19 (1983) and
Zeit. für W-theorie 65 (1984). Characterizating of manifold-valued martingales by convex functions has become a powerful tool: see for instance Émery's book
Stochastic Calculus in Manifolds (Springer, 1989) and his St-Flour lectures (Springer LNM 1738)
Keywords: Martingales in manifolds,
Semimartingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
XVII: 09, 89-105, LNM 986 (1983)
YOR, Marc
Le drap brownien comme limite en loi des temps locaux linéaires (
Brownian motion,
Local time,
Brownian sheet)
A central limit theorem is obtained for the increments $L^x_t-L^0_t$ of Brownian local times. The limiting process is expressed in terms of a Brownian sheet, independent of the initial Brownian motion
Comment: This type of result is closely related to the Ray-Knight theorems, which describe the law of Brownian local times considered at certain random times. This has been extended first by Rosen in
2533, where Brownian motion is replaced with a symmetric stable process, then by Eisenbaum
2926Keywords: Brownian motion,
Several parameter processesNature: Original Retrieve article from Numdam
XVII: 18, 179-184, LNM 986 (1983)
HE, Sheng-Wu;
YAN, Jia-An;
ZHENG, Wei-An
Sur la convergence des semimartingales continues dans ${\bf R}^n$ et des martingales dans une variété (
Stochastic calculus,
Stochastic differential geometry)
Say that a continuous semimartingale $X$ with canonical decomposition $X_0+M+A$ converges perfectly on an event $E$ if both $M_t$ and $\int_0^t|dA_s|$ have an a.s. limit on $E$ when $t\rightarrow \infty $. It is established that if $A_t$ has the form $\int_0^tH_sd[M,M]_s$, $X$ converges perfectly on the event $\{\sup_t|X_t|+\lim\sup_tH_t <\infty \}$. A similar (but less simple) statement is shown for multidimensional $X$; and an application is given to martingales in manifolds: every point of a manifold $V$ (with a connection) has a neighbourhood $U$ such that, given any $V$-valued martingale $X$, almost all paths of $X$ that eventually remain in $U$ are convergent
Comment: The latter statement (martingale convergence) is very useful; more recent proofs use convex functions instead of perfect convergence. The next talk
1719 is a small remark on perfect convergence
Keywords: Semimartingales,
Martingales in manifoldsNature: Original Retrieve article from Numdam
XVII: 19, 185-186, LNM 986 (1983)
ÉMERY, Michel
Note sur l'exposé précédent (
Stochastic calculus)
A small remark on
1718: The event where a semimartingale converges perfectly is also the smallest (modulo negligibility) event where it is a semimartingale up to infinity
Keywords: SemimartingalesNature: Original additions Retrieve article from Numdam
XVII: 21, 194-197, LNM 986 (1983)
PRICE, Gareth C.;
WILLIAMS, David
Rolling with `slipping': I (
Stochastic calculus,
Stochastic differential geometry)
If $Z$ and $\tilde Z$ are two Brownian motions on the unit sphere for the filtration of $Z$, there differentials $\partial Y=(\partial Z) \times Z$ (Stratonovich differentials and vector product) and $\partial\tilde Y$ (similarly defined) are related by $d\tilde Y = H dY$, where $H$ is a previsible, orthogonal transformation such that $HZ=\tilde Z$
Keywords: Brownian motion in a manifold,
Previsible representationNature: Original Retrieve article from Numdam
XVII: 22, 198-204, LNM 986 (1983)
KARANDIKAR, Rajeeva L.
Girsanov type formula for a Lie group valued Brownian motion (
Brownian motion,
Stochastic differential geometry)
A formula for the change of measure of a Lie group valued Brownian motion is stated and proved. It needs a Borel correspondence between paths in the Lie algebra and paths in the group, that transforms all (continuous) semimartingales in the algebra into their stochastic exponential
Comment: For more on stochastic exponentials in Lie groups, see Hakim-Dowek-Lépingle
2023 and Arnaudon
2612Keywords: Changes of measure,
Brownian motion in a manifold,
Lie groupNature: Original Retrieve article from Numdam
XVII: 24, 221-224, LNM 986 (1983)
BASS, Richard F.
Skorohod imbedding via stochastic integrals (
Brownian motion)
A centered probability $\mu$ on $\bf R$ is the law of $g(X_1)$, for a suitable function $g$ and $(X_t,\ t\le 1)$ a Brownian motion. The martingale with terminal value $g(X_1)$ is a time change $(T(t), \ t\le1)$ of a Brownian motion $\beta$; it is shown that $T(1)$ is a stopping time for $\beta$, thus showing the Skorohod embedding for $\mu$
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XVII: 25, 225-226, LNM 986 (1983)
MEILIJSON, Isaac
On the Azéma-Yor stopping time (
Brownian motion)
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbeddingNature: Original Retrieve article from Numdam
XVII: 26, 227-239, LNM 986 (1983)
VALLOIS, Pierre
Le problème de Skorokhod sur $\bf R$ : une approche avec le temps local (
Brownian motion)
A solution to Skorohod's embedding problem is given, using the first hiting time of a set by the 2-dimensional process which consists of Brownian motion and its local time at zero. The author's aim is to ``correct'' the asymmetry inherent to the Azéma-Yor construction
Comment: A general survey on the Skorohod embedding problem is Ob\lój,
Probab. Surv. 1, 2004
Keywords: Skorohod imbedding,
Local timesNature: Original Retrieve article from Numdam
XVII: 28, 243-297, LNM 986 (1983)
ALDOUS, David J.
Random walks on finite groups and rapidly mixing Markov chains (
Markov processes)
The ``mixing time'' for a Markov chain---how many steps are needed to approximately reach the stationary distribution---is defined here by taking the variation distance between measures and the worst possible starting point, and bounded above by coupling arguments. For the simple random walk on the discrete cube $\{0,1\}^d$ with large $d$, there is a ``cut-off phenomenon'', an abrupt change in variation distance from 1 to 0 around time $1/4\ d\,\log d$. For a natural model of riffle shuffle of an $n$-card deck, there is an analogous cut-off at time $3/2\ \log n$. The relationship between ``rapid mixing'' and approximate exponential distribution for first hitting times on small subsets, is also discussed
Comment: In the 1960s and 1970s, Markov chains were considered by probabilists as rather trite objects. This work was one of several papers that prompted a reassessment and focused attention on the question of mixing time. In 1981, Diaconis-Sashahani (
Z. Wahrsch. Verw. Gebiete 57) had established the cut-off phenomenon for a different shuffling scheme. For a random walk on a graph, Alon (
Combinatorica 6, 1986) related an eigenvalue-based mixing time to expansion properties of the graph, and parallel work of Lawler-Sokal (
Trans. Amer. Math. Soc. 309, 1988) in the broader setting of reversible chains made a connection with models from statistical physics. Jerrum-Sinclair (
Inform. and Comput. 82, 1989) gave the first deep use of Markov chain methods in the theory of algorithms, while Geman-Geman (
IEEE Trans. Pattern Anal. Machine Intell. 6, 1984) promoted the use of Markov chains in image reconstruction. Such papers brought the attention of probabilists to the Metropolis algorithm in statistical physics, and foreshadowed the development of Markov chain Monte Carlo methods in Bayesian statistics, e.g. Smith (
Philos. Trans. Roy. Soc. London 337, 1991)
Keywords: Markov chains,
Hitting probabilitiesNature: Original Retrieve article from Numdam
XVIII: 32, 500-500, LNM 1059 (1984)
ÉMERY, Michel
Sur l'exponentielle d'une martingale de $BMO$ (
Martingale theory)
This very short note remarks that for complex-valued processes, it is no longer true that the stochastic exponential of a bounded martingale is a martingale---it is only a local martingale
Keywords: Stochastic exponentials,
$BMO$Nature: Original Retrieve article from Numdam
XVIII: 33, 501-518, LNM 1059 (1984)
ÉMERY, Michel;
ZHENG, Wei-An
Fonctions convexes et semimartingales dans une variété (
Stochastic differential geometry)
On a manifold endowed with a connexion, convex functions can be defined, and transform manifold-valued martingales into real-valued local submartingales (see Darling
1659). This is extended here to the case of non-smooth convex functions. Ii is also shown that they make manifold-valued semimartingales into real semimartingales
Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
XIX: 07, 91-112, LNM 1123 (1985)
SCHWARTZ, Laurent
Construction directe d'une diffusion sur une variété (
Stochastic differential geometry)
This seems to be the first use of Witney's embedding theorem to construct a process (a Brownian motion, a diffusion, a solution to some s.d.e.) in a manifold $M$ by embedding $M$ into some $
R^d$. Very general existence and uniqueness results are obtained
Comment: This method has since become standard in stochastic differential geometry; see for instance Émery's book
Stochastic Calculus in Manifolds (Springer, 1989)
Keywords: Diffusions in manifolds,
Stochastic differential equationsNature: Original Retrieve article from Numdam
XIX: 22, 271-274, LNM 1123 (1985)
LÉANDRE, Rémi
Flot d'une équation différentielle stochastique avec semimartingale directrice discontinue (
Stochastic calculus)
Given a good s.d.e. of the form $dX=F\circ X_- dZ$, $X_{t-}$ is obtained from $X_t$ by computing $H_z(x) = x+F(x)z$, where $z$ stands for the jump of $Z$. Call $D$ (resp. $I$ the set of all $z$ such that $H_z$ is a diffeomorphism (resp. injective). It is shown that the flow associated to the s.d.e. is made of diffeomorphisms (respectively is one-to-one) iff all jumps of $Z$ belong to $D$ (resp. $I$)
Keywords: Stochastic differential equations,
Flow of a s.d.e.Nature: Original Retrieve article from Numdam
XIX: 27, 297-313, LNM 1123 (1985)
LE GALL, Jean-François
Sur la mesure de Hausdorff de la courbe brownienne (
Brownian motion)
Previous results on the $h$-measure of the Brownian curve in $
R^2$ or $
R^3$ indexed by $t\in[0,1]$, by Cisielski-Taylor
Trans. Amer. Math. Soc. 103 (1962) and Taylor
Proc. Cambridge Philos. Soc. 60 (1964) are sharpened. The method uses the description à la Ray-Knight of the local times of Bessel processes
Comment: These Ray-Knight descriptions are useful ; they were later used in questions not related to Hausdorff measures. See for instance Biane-Yor,
Ann. I.H.P. 23 (1987), Yor,
Ann. I.H.P. 27 (1991)
Keywords: Hausdorff measures,
Brownian motion,
Bessel processes,
Ray-Knight theoremsNature: Original Retrieve article from Numdam
XIX: 28, 314-331, LNM 1123 (1985)
LE GALL, Jean-François
Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan (
Brownian motion)
The normalized self-intersection local time of planar Brownian motion was shown to exist by Varadhan (Appendix to
Euclidean quantum field theory, by K.~Symanzik, in
Local Quantum Theory, Academic Press, 1969). This is established anew here by a completely different method, using the intersection local time of two independent planar Brownian motions (whose existence was established by Geman, Horowitz and Rosen,
Ann. Prob. 12, 1984) and a sequence of dyadic decompositions of the triangle $\{0<s<t\le1\}$
Comment: Later, Dynkin, Rosen, Le Gall and others have shown existence of a renormalized local time for the multiple self-intersection of arbitrary order $n$ of planar Brownian motion. A good reference is Le Gall,
École d'Été de Saint-Flour XX, Springer LNM 1527
Keywords: Brownian motion,
Local times,
Self-intersectionNature: Original proofs Retrieve article from Numdam
XIX: 29, 332-349, LNM 1123 (1985)
YOR, Marc
Compléments aux formules de Tanaka-Rosen (
Brownian motion)
Several variants of Rosen's works (
Comm. Math. Phys. 88 (1983),
Ann. Proba. 13 (1985),
Ann. Proba. 14 (1986)) are presented. They yield Tanaka-type formulae for the self-intersection local times of Brownian motion in dimension 2 and beyond, establishing again Varadhan's normalization result (Appendix to
Euclidean quantum field theory, by K.~Symanzik, in
Local Quantum Theory, Academic Press, 1969). The methods involve stochastic calculus, which was not needed in
1928Comment: Examples of further work on this subject, using stochastic calculus or not, are Werner,
Ann. I.H.P. 29 (1993) who gives many references, Khoshnevisan-Bass,
Ann. I.H.P. 29 (1993), Rosen-Yor
Ann. Proba. 19 (1991)
Keywords: Brownian motion,
Local times,
Self-intersectionNature: Original proofs Retrieve article from Numdam
XIX: 30, 350-365, LNM 1123 (1985)
YOR, Marc
Renormalisation et convergence en loi pour des temps locaux d'intersection du mouvement brownien dans ${\bf R}^3$ (
Brownian motion)
It is shown that no renormalization à la Varadhan occurs for the self-intersection local times of 3-dimensional Brownian motion; but a weaker result is established: when the point $y\in
R^3$ tends to $0$, the self-intersection local time at $y$, on the triangle $\{0<s<u\le t\},\ t\ge0$, centered and divided by $(-\log|y|)^{1/2}$, converges in law to a Brownian motion. Several variants of this theorem are established
Comment: This result was used by Le Gall in his work on fluctuations of the Wiener sausage:
Ann. Prob. 16 (1988). Many results by Rosen have the same flavour
Keywords: Brownian motion,
Local times,
Self-intersectionNature: Original Retrieve article from Numdam
XX: 12, 131-161, LNM 1204 (1986)
BOULEAU, Nicolas;
HIRSCH, Francis
Propriété d'absolue continuité dans les espaces de Dirichlet et applications aux équations différentielles stochastiques (
Dirichlet forms,
Malliavin's calculus)
This is the main result of the ``Bouleau-Hirsch approach'' to absolute continuity in Malliavin calculus (see
The Malliavin calculus and related topics by D. Nualart, Springer1995). In the framework of Dirichlet spaces, a general criterion for absolute continuity of random vectors is established; it involves the image of the energy measure. This leads to a Lipschitzian functional calculus for the Ornstein-Uhlenbeck Dirichlet form on Wiener space, and gives absolute continuity of the laws of the solutions to some SDE's with coefficients that can be uniformly degenerate
Comment: These results are extended by the same authors in their book
Dirichlet Forms and Analysis on Wiener Space, De Gruyter 1991
Keywords: Dirichlet forms,
Carré du champ,
Absolute continuity of lawsNature: Original Retrieve article from Numdam
XX: 13, 162-185, LNM 1204 (1986)
BOULEAU, Nicolas;
LAMBERTON, Damien
Théorie de Littlewood-Paley et processus stables (
Applications of martingale theory,
Markov processes)
Meyer' probabilistic approach to Littlewood-Paley inequalities (
1010,
1510) is extended by replacing the underlying Brownian motion with a stable process. The following spectral multiplicator theorem is obtained: If $(P_t)_{t\geq 0}$ is a symmetric Markov semigroup with spectral representation $P_t=\int_{[0,\infty)}e^{-t\lambda} dE_{\lambda}$, and if $M$ is a function on $
R_+$ defined by $M(\lambda)=\lambda\int_0^\infty r(y)e^{-y\lambda}dy,$ where $r(y)$ is bounded and Borel on $
R_+$, then the operator $T_M=\int_{[0,\infty)}M(\lambda)dE_{\lambda},$ which is obviously bounded on $L^2$, is actually bounded on all $L^p$ spaces of the invariant measure, $1<p<\infty$. The method also leads to new Littlewood-Paley inequalities for semigroups admitting a carré du champ operator
Keywords: Littlewood-Paley theory,
Semigroup theory,
Riesz transforms,
Stable processes,
Inequalities,
Singular integrals,
Carré du champNature: Original Retrieve article from Numdam
XX: 23, 352-374, LNM 1204 (1986)
HAKIM-DOWEK, M.;
LÉPINGLE, Dominique
L'exponentielle stochastique des groupes de Lie (
Stochastic differential geometry)
Given a Lie group $G$ and its Lie algebra $\cal G$, this article defines and studies the stochastic exponential of a (continuous) semimartingale $M$ in $\cal G$ as the solution in $G$ to the Stratonovich s.d.e. $dX = X dM$. The inverse operation (stochastic logarithm) is also considered; various formulas are established (e.g. the exponential of $M+N$). When $M$ is a local martingale, $X$ is a martingale for the connection such that $\nabla_A B=0$ for all left-invariant vector fields $A$ and $B$
Comment: See also Karandikar
Ann. Prob. 10 (1982) and
1722. For a sequel, see Arnaudon
2612Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Lie groupNature: Original Retrieve article from Numdam
XX: 31, 465-502, LNM 1204 (1986)
McGILL, Paul
Integral representation of martingales in the Brownian excursion filtration (
Brownian motion,
Stochastic calculus)
An integral representation is obtained of all square integrable martingales in the filtration $({\cal E}^x,\ x\in
R)$, where ${\cal E}^x$ denotes the Brownian excursion $\sigma$-field below $x$ introduced by D. Williams
1343, who also showed that every $({\cal E}^x)$ martingale is continuous
Comment: Another filtration $(\tilde{\cal E}^x,\ x\in
R)$ of Brownian excursions below $x$ has been proposed by Azéma; the structure of martingales is quite diffferent: they are discontinuous. See Y. Hu's thesis (Paris VI, 1996), and chap.~16 of Yor,
Some Aspects of Brownian Motion, Part~II, Birkhäuser, 1997
Keywords: Previsible representation,
Martingales,
FiltrationsNature: Original Retrieve article from Numdam
XX: 33, 515-531, LNM 1204 (1986)
ROSEN, Jay S.
A renormalized local time for multiple intersections of planar Brownian motion (
Brownian motion)
Using Fourier techniques, the existence of a renormalized local time for $n$-fold self-intersections of planar Brownian motion is obtained, thus extending the case $n=2$, obtained in the pioneering work of Varadhan (Appendix to
Euclidean quantum field theory, by K.~Symanzik, in
Local Quantum Theory, Academic Press, 1969)
Comment: Closely related to
2036. A general reference is Le Gall,
École d'Été de Saint-Flour XX, Springer LNM 1527
Keywords: Local times,
Self-intersectionNature: Original Retrieve article from Numdam
XX: 35, 543-552, LNM 1204 (1986)
YOR, Marc
Sur la représentation comme intégrales stochastiques des temps d'occupation du mouvement brownien dans ${\bf R}^d$ (
Brownian motion)
Varadhan's renormalization result (Appendix to
Euclidean quantum field theory, by K.~Symanzik, in
Local Quantum Theory consists in centering certain sequences of Brownian functionals and showing $L^2$-convergence. The same results are obtained here by writing these centered functionals as stochastic integrals
Comment: One of mny applications of stochastic calculus to the existence and regularity of self-intersection local times. See Rosen's papers on this topic in general, and page 196 of Le Gall,
École d'Été de Saint-Flour XX, Springer LNM 1527
Keywords: Local times,
Self-intersection,
Previsible representationNature: Original proofs Retrieve article from Numdam
XXIV: 28, 407-441, LNM 1426 (1990)
ÉMERY, Michel
On two transfer principles in stochastic differential geometry (
Stochastic differential geometry)
Second-order stochastic calculus gives two intrinsic methods to transform an ordinary differential equation into a stochastic one (see Meyer
1657, Schwartz
1655 or Emery
Stochastic calculus in manifolds). The first one gives a Stratonovich SDE and needs coefficients regular enough; the second one gives an Ito equation and needs a connection on the manifold. Discretizing time and smoothly interpolating the driving semimartingale is known to give an approximation to the Stratonovich transfer; it is shown here that another discretized-time procedure converges to the Ito transfer. As an application, if the ODE makes geodesics to geodesics, then the Ito and Stratonovich SDE's are the same
Comment: An error is corrected in
2649. The term ``transfer principle'' was coined by Malliavin,
Géométrie Différentielle Stochastique, Presses de l'Université de Montréal (1978); see also Bismut,
Principes de Mécanique Aléatoire (1981) and
1505Keywords: Stochastic differential equations,
Semimartingales in manifolds,
Transfer principleNature: Original Retrieve article from Numdam
XXIV: 30, 448-452, LNM 1426 (1990)
ÉMERY, Michel;
LÉANDRE, Rémi
Sur une formule de Bismut (
Markov processes,
Stochastic differential geometry)
This note explains why, in Bismut's work on the index theorem, the reference measure is not the Riemannian measure $r$ on the manifold, but $p_1(x,x) r(dx)$, where $p_t(x,y)$ is the density (with respect to $r$!) of the Brownian semi-group
Keywords: Brownian bridge,
Brownian motion in a manifold,
Transformations of Markov processesNature: Exposition,
Original additions Retrieve article from Numdam
XXV: 18, 196-219, LNM 1485 (1991)
PICARD, Jean
Calcul stochastique avec sauts sur une variété (
Stochastic differential geometry)
It is known from Meyer
1505 that intrinsic Ito integrals have a meaning for continuous semimartingales in a manifold $M$, provided $M$ is endowed with a connection. This is extended here to càdlàg semimartingales. The manifold must be endowed with a richer structure, a ``connector'', mapping $M\times M$ to the tangent bundle, that allows to interpret a jump $(X_{t-},X_t)$ as a tangent vector to $M$ at $X{t-}$; the differential of the connector at the diagonal reduces to a classical torsion-free connection. Introducing torsions leads to a more general ``transporter'', describing how parallel transports should behave at jump times, and reducing to a classical connection for infinitesimal jumps. Discrete-time approximations are established.
Keywords: Semimartingales in manifolds,
Martingales in manifolds,
JumpsNature: Original Retrieve article from Numdam
XXV: 19, 220-233, LNM 1485 (1991)
ÉMERY, Michel;
MOKOBODZKI, Gabriel
Sur le barycentre d'une probabilité dans une variété (
Stochastic differential geometry)
In a manifold $V$ (endowed with a connection), convex functions and continuous martingales can be defined, but expectations cannot. This article proposes to define the mass-centre of a probability $\mu$ on $V$ as a whole set of points, consisting of all $x$ in $V$ such that $f(x)\le\mu(f)$ for all bounded, convex $f$ on $V$. If $V$ is small enough, it is shown that this is equivalent to demanding that there exists (on a suitable filtered probability space) a continuous martingale $X$ such that $X_0=x$ and $X_1$ has law $\mu$
Comment: The conjecture (due to Émery) at the bottom of page 232 has been disproved by Kendall (
J. London Math. Soc. 46, 1992), as pointed out in
2650Keywords: Martingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
XXV: 33, 407-424, LNM 1485 (1991)
ROSEN, Jay S.
Second order limit laws for the local times of stable processes (
Limit theorems)
Using the method of moments, a central limit theorem is established for the increments $L^x_t-L^0_t$ of the local times of a symmetric $\beta$-stable process ($\beta>1$). The limit law is that of a fractional Brownian sheet, with Hurst index $\beta-1$, time-changed via $L_t^0$ in its time variable
Comment: Another proof due to Eisenbaum
3120 uses Dynkin's isomorphism. Ray-Knight theorems for these local times can be found in Eisenbaum-Kaspi-Marcus-Rosen-Shi
Ann. Prob. 28 (2000). A good reference on this subject is Marcus-Rosen,
Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006)
Keywords: Local times,
Stable processes,
Method of moments,
Fractional Brownian motion,
Brownian sheetNature: Original Retrieve article from Numdam
XXVI: 10, 113-126, LNM 1526 (1992)
TAYLOR, John C.
Skew products, regular conditional probabilities and stochastic differential equations: a technical remark (
Stochastic calculus,
Stochastic differential geometry)
This is a detailed study of the transfer principle (the solution to a Stratonovich stochastic differential equations can be pathwise obtained from the driving semimartingale by solving the corresponding ordinary differential equation) in the case of an equation where the solution of another equation plays the role of a parameter
Comment: The term ``transfer principle'' was coined by Malliavin,
Géométrie Différentielle Stochastique, Presses de l'Université de Montréal (1978); see also Bismut,
Principes de Mécanique Aléatoire (1981)
Keywords: Transfer principle,
Stochastic differential equations,
Stratonovich integralsNature: Original Retrieve article from Numdam
XXVI: 11, 127-145, LNM 1526 (1992)
ESTRADE, Anne;
PONTIER, Monique
Relèvement horizontal d'une semimartingale càdlàg (
Stochastic differential geometry,
Stochastic calculus)
For filtering purposes, the lifting of a manifold-valued semimartingale $X$ to the tangent space at $X_0$ is extended here to the case when $X$ has jumps. The value of $L_t$ involves the inverse of the exponential at $X_{t-}$ applied to $X_t$, and a parallel transport from $X_0$ to $X_{t-}$
Comment: The same method is described in a more general setting by Kurtz-Pardoux-Protter
Ann I.H.P. (1995). In turn, this is a particular instance of a very general scheme due to Cohen (
Stochastics Stoch. Rep. (1996)
Keywords: Stochastic parallel transport,
Stochastic differential equations,
JumpsNature: Original Retrieve article from Numdam
XXVI: 12, 146-154, LNM 1526 (1992)
ARNAUDON, Marc
Connexions et martingales dans les groupes de Lie (
Stochastic differential geometry)
The stochastic exponential of Hakim-Dowek-Lépingle
2023 is interpreted in terms of second-order geometry, studied in details and generalized
Keywords: Martingales in manifolds,
Lie groupNature: Original Retrieve article from Numdam
XXVI: 13, 155-156, LNM 1526 (1992)
ARNAUDON, Marc;
MATTHIEU, Pierre
Appendice : Décomposition en produit de deux browniens d'une martingale à valeurs dans un groupe muni d'une métrique bi-invariante (
Stochastic differential geometry)
Using
2612, it is shown that in a Lie group with a bi-invariant Riemannian structure, every martingale is a time-changed product of two Brownian motions
Keywords: Martingales in manifolds,
Lie groupNature: Original Retrieve article from Numdam
XXVI: 18, 189-209, LNM 1526 (1992)
NORRIS, James R.
A complete differential formalism for stochastic calculus in manifolds (
Stochastic differential geometry)
The use of equivariant coordinates in stochastic differential geometry is replaced here by an equivalent, but intrinsic, formalism, where the differential of a semimartingale lives in the tangent bundle. Simple, intrinsic Girsanov and Feynman-Kac formulas are given, as well as a nice construction of a Brownian motion in a manifold admitting a Riemannian submersion with totally geodesic fibres
Keywords: Semimartingales in manifolds,
Stochastic integrals,
Feynman-Kac formula,
Changes of measure,
Heat semigroupNature: Original Retrieve article from Numdam
XXVI: 24, 322-347, LNM 1526 (1992)
JEULIN, Thierry;
YOR, Marc
Une décomposition non-canonique du drap brownien (
Brownian sheet,
Gaussian processes)
In
2415, the authors have introduced a transform of Brownian motion. Here, a similar transform is defined on the Brownian sheet; this transform is shown to be strongly mixing
Comment: This work was motivated by Föllmer's article on Martin boundaries on Wiener space (in
Diffusion processes and related problems in analysis, vol.~I, Birkhäuser 1990)
Keywords: Brownian motion,
Several parameter processesNature: Original Retrieve article from Numdam
XXVII: 14, 122-132, LNM 1557 (1993)
DUBINS, Lester E.;
ÉMERY, Michel;
YOR, Marc
On the Lévy transformation of Brownian motions and continuous martingalesComment: An erratum is given in
4421 in Volume XLIV.
Nature: Original Retrieve article from Numdam
XXIX: 16, 166-180, LNM 1613 (1995)
APPLEBAUM, David
A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (
Stochastic differential geometry,
Markov processes)
This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.
Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (
Stochastics Stochastics Rep. 56, 1996). The same question is addressed by Cohen in the next article
2917Keywords: Semimartingales with jumps,
Lévy processes,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
XXIX: 17, 181-193, LNM 1613 (1995)
COHEN, Serge
Some Markov properties of stochastic differential equations with jumps (
Stochastic differential geometry,
Markov processes)
The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see
1505 and
1655) was extended by Cohen to càdlàg semimartingales (
Stochastics Stochastics Rep. 56, 1996). Here this language is used to study the Markov property of solutions to SDE's with jumps. In particular,two definitions of a Lévy process in a Riemannian manifold are compared: One as the solution to a SDE driven by some Euclidean Lévy process, the other by subordinating some Riemannian Brownian motion. It is shown that in general the former is not of the second kind
Comment: The first definition is independently introduced by David Applebaum
2916Keywords: Semimartingales with jumps,
Lévy processes,
Subordination,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
XXIX: 18, 194-201, LNM 1613 (1995)
FRANCHI, Jacques
Chaos multiplicatif : un traitement simple et complet de la fonction de partition (
Statistical mechanics)
Introduced by Mandelbrot (
Comptes Rendus Acad. Sci. 278, 289--292, 1974), the model of multiplicative chaos has since been studied by several mathematicians and physicists. Using a trick of Kahane, this article presents a complete and elementary calculation of the pressure, thereby completing and simplifying previous work by Collet and Koukiou. Moreover it connects the critical temperature to the entropy, and gives a necessary and sufficient condition for finiteness of the critical temperature
Keywords: Multiplicative chaos,
Partition function,
Pressure,
Critical temperatureNature: Original Retrieve article from Numdam
XXIX: 26, 266-289, LNM 1613 (1995)
EISENBAUM, Nathalie
Une version sans conditionnement du théorème d'isomorphisme de Dynkin (
Limit theorems)
After establishing an unconditional version of Dynkin's isomorphism theorem, the author applies this theorem to give a new proof of Ray-Knight theorems for Brownian local times, and also to give another proof to limit theorems due to Rosen
2533 concerning the increments of the local times of a symmetric $\beta$-stable process for $\beta>1$. Some results by Marcus-Rosen (
Proc. Conf. Probability in Banach Spaces~8, Birkhäuser 1992) on Laplace transforms of the increments of local time are extended
Comment: A general reference on the subject is Marcus-Rosen,
Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006)
Keywords: Stable processes,
Local times,
Central limit theorem,
Dynkin isomorphism,
Fractional Brownian motion,
Brownian sheetNature: Original Retrieve article from Numdam
XXXI: 05, 54-61, LNM 1655 (1997)
FANG, Shizan;
FRANCHI, Jacques
A differentiable isomorphism between Wiener space and path group (
Malliavin's calculus)
The Itô map $I$ is known to realize a measurable isomorphism between Wiener space $W$ and the group ${\cal P}$ of paths with values in a Riemannian manifold. Here, the pullback $I^{*}$ is shown to be a diffeomorphism (in the sense of Malliavin derivatives) between the exterior algebras $\Lambda (W)$ and $\Lambda ({\cal P})$. This allows to transfer the Weitzenböck-Shigekawa identity from $\Lambda (W)$ to $\Lambda ({\cal P})$, yielding for example the de~Rham-Hodge-Kodaira decomposition on ${\cal P}$
Keywords: Wiener space,
Path group,
Brownian motion in a manifold,
Differential formsNature: Original Retrieve article from Numdam
XXXI: 12, 113-125, LNM 1655 (1997)
ELWORTHY, Kenneth David;
LI, Xu-Mei;
YOR, Marc
On the tails of the supremum and the quadratic variation of strictly local martingales (
Martingale theory)
The asymptotic tails of the current maximum and the quadratic variation of a positive continuous local martingale are compared. Applications to strict local martingales associated with transient diffusions, such as Bessel processes, and remarkable identities for Bessel functions are given
Comment: In discrete time, see the following article
3113. Related results are due to Takaoka
3313Keywords: Continuous martingales,
Local martingales,
Quadratic variation,
Maximal processNature: Original Retrieve article from Numdam
XXXI: 16, 176-189, LNM 1655 (1997)
ÉMERY, Michel
Closed sets supporting a continuous divergent martingale (
Martingale theory)
This note gives a characterization of all closed subsets $F$ of $
R^d$ such that, for every $F$-valued continuous martingale $X$, the limit $X_\infty$ exists in $F$ (or $
R^d$) with non-zero probability. The criterion is as follows: To each $F$ is associated a smaller closed set $F'$ obtained, roughly speaking, by chopping off all prominent parts of $F$; this map $F\mapsto F'$ is iterated, giving a decreasing sequence $(F^n)$ with limit $F^\infty$; the condition is that $F^\infty$ is empty. (If $d=2$, $F^\infty$ is also the largest closed subset of $F$ such that all connected components of its complementary are convex)
Comment: Two similar problems are discussed in
1485Keywords: Continuous martingales,
Asymptotic behaviour of processesNature: Original Retrieve article from Numdam
XXXI: 20, 216-224, LNM 1655 (1997)
EISENBAUM, Nathalie
Théorèmes limites pour les temps locaux d'un processus stable symétrique (
Limit theorems)
Using Dynkin's isomorphism, a central-limit type theorem is derived for the local times of a stable symmetric process of index $\beta$ at a finite number $n$ of levels. The limiting process is expressed in terms of a fractional, $n$-dimensional Brownian sheet with Hurst index $\beta-1$. The case when $n=1$ is due to Rosen
2533, and, for Brownian local times, to Yor
1709Comment: This kind of result is now understood as a weak form of theorems à la Ray-Knight, describing the local times of a stable symmetric process: see Eisenbaum-Kaspi-Marcus-Rosen-Shi
Ann. Prob. 28 (2000) for a Ray-Knight theorem involving fractional Brownian motion. Marcus-Rosen,
Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press (2006) is a general reference on the subject
Keywords: Stable processes,
Local times,
Central limit theorem,
Dynkin isomorphism,
Fractional Brownian motion,
Brownian sheetNature: Original Retrieve article from Numdam
XXXI: 25, 256-265, LNM 1655 (1997)
TAKAOKA, Koichiro
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman's theorem (
Stochastic calculus)
Martingales involving the future minimum of a transient Bessel process are studied, and shown to satisfy a non Markovian SDE. In dimension $>3$, uniqueness in law does not hold for this SDE. This generalizes Saisho-Tanemura
Tokyo J. Math. 13 (1990)
Comment: Extended to more general diffusions in the next article
3126Keywords: Continuous martingales,
Bessel processes,
Pitman's theoremNature: Original Retrieve article from Numdam
XXXI: 26, 266-271, LNM 1655 (1997)
RAUSCHER, Bernhard
Some remarks on Pitman's theorem (
Stochastic calculus)
For certain transient diffusions $X$, local martingales which are functins of $X_t$ and the future infimum $\inf_{u\ge t}X_u$ are constructed. This extends the preceding article
3125Comment: See also chap. 12 of Yor,
Some Aspects of Brownian Motion Part~II, Birkhäuser (1997)
Keywords: Continuous martingales,
Bessel processes,
Diffusion processes,
Pitman's theoremNature: Original Retrieve article from Numdam
XXXI: 29, 306-314, LNM 1655 (1997)
YOR, Marc
Some remarks about the joint law of Brownian motion and its supremum (
Brownian motion)
Seshadri's identity says that if $S_1$ denotes the maximum of a Brownian motion $B$ on the interval $[0,1]$, the r.v. $2S_1(S_1-B_1)$ is independent of $B_1$ and exponentially distributed. Several variants of this are obtained
Comment: See also
3320Keywords: Maximal process,
Seshadri's identityNature: Original Retrieve article from Numdam
XXXII: 19, 264-305, LNM 1686 (1998)
BARLOW, Martin T.;
ÉMERY, Michel;
KNIGHT, Frank B.;
SONG, Shiqi;
YOR, Marc
Autour d'un théorème de Tsirelson sur des filtrations browniennes et non browniennes (
Brownian motion,
Filtrations)
Tsirelson has shown that no Walsh's Brownian motion with three rays or more can live in a Brownian filtration (GAFA
7, 1997). Using his methods, the result is extended to spider martingales. A conjecture of M. Barlow is also proved: if $L$ is an honest time in a (possibly multidimensional) Brownian filtration, then ${\cal F}_{L+}$ is generated by ${\cal F}_{L}$ and at most one event. Last, it is shown that a Walsh's Brownian motion can live in the filtration generated by another Walsh's Brownian motion only if the former is obtained from the latter by aggregating rays
Comment: On Tsirelson's theorem, see also Tsirelson, ICM 1998 vol. III, and M. Émery,
Astérisque 282 (2002). A simplified proof of Barlow's conjecture is given in
3304. For more on Théorème 1 (Slutsky's lemma), see
3221 and
3325Keywords: Filtrations,
Spider martingales,
Walsh's Brownian motion,
Cosiness,
Slutsky's lemmaNature: New exposition of known results,
Original additions Retrieve article from Numdam
XXXIII: 13, 327-333, LNM 1709 (1999)
TAKAOKA, Koichiro
Some remarks on the uniform integrability of continuous martingales (
Martingale theory)
For a continuous local martingale which converges a.s., a general relation links the asymptotic tails of the maximal variable and the quadratic variation. This unifies previous results by Azéma-Gundy-Yor
1406, Elworthy-Li-Yor
3112 and
Probab. Theory Related Fields 115 (1999)
Keywords: Uniform integrability,
Continuous martingales,
Local martingalesNature: Original Retrieve article from Numdam
XXXIII: 16, 342-348, LNM 1709 (1999)
GRANDITS, Peter
Some remarks on L$^\infty $, H$^\infty $, and $BMO$ (
Martingale theory)
It is known from
1212 that neither $L^\infty$ nor $H^\infty$ is dense in $BMO$. This article answers a question raised by Durrett (
Brownian Motion and Martingales in Analysis, Wadworth 1984): Does there exist a $BMO$-martingale which has a best approximation in $L^\infty$? The answer is negative, but becomes positive if $L^\infty$ is replaced with $H^\infty$
Keywords: $BMO$,
Hardy spacesNature: Original Retrieve article from Numdam
XXXIII: 20, 388-394, LNM 1709 (1999)
PITMAN, James W.
The distribution of local times of a Brownian bridge (
Brownian motion)
Several useful identities for the one-dimensional marginals of local times of Brownian bridges are derived. This is a variation and extension on the well-known joint law of the maximum and the value of Brownian motion at a given time
Comment: Useful references are Borodin,
Russian Math. Surveys (1989) and the book
Brownian motion and stochastic calculus by Karatzas-Shrieve (Springer, 1991)
Keywords: Local times,
Brownian bridgeNature: Original Retrieve article from Numdam
XLI: 01, 1-18, LNM 1934 (2008)
DERMOUNE, Azzouz;
HEINRICH, Philippe
Spectral gap inequality for a colored disordered lattice gasNature: Original
XLI: 02, 19-49, LNM 1934 (2008)
FÉRAL, D.
On large deviations for the spectral measure of discrete Coulomb gasNature: Original
XLI: 03, 51-92, LNM 1934 (2008)
KHORUNZHIY, Oleksiy
Estimates for moments of random matrices with Gaussian elements (
Random matrices)
Nature: Original
XLI: 04, 93-119, LNM 1934 (2008)
CAPITAINE, M.;
CASALIS, M.
Geometric interpretation of the cumulants for random matrices previously defined as convolutions on the symmetric group (
Random matrices)
Nature: Original
XLI: 05, 121-135, LNM 1934 (2008)
KYPRIANOU, Andreas E.;
PALMOWSKI, Zbigniew
Fluctuations of spectrally negative Markov additive processes (
Theory of Markov processes)
Nature: Original
XLI: 06, 137-159, LNM 1934 (2008)
BERTOIN, Jean;
LINDNER, Alexander;
MALLER, Ross
On continuity properties of the law of integrals of Lévy processes (
Theory of Lévy processes)
Nature: Original
XLI: 07, 161-179, LNM 1934 (2008)
BARAKA, Driss;
MOUNTFORD, Thomas
A law of the iterated logarithm for fractional Brownian motions (
Theory of fractional Brownian motion)
Nature: Original
XLI: 08, 181-197, LNM 1934 (2008)
NOURDIN, Ivan
A simple theory for the study of SDEs driven by a fractional Brownian motion in dimension one (
Theory of fractional Brownian motion)
Nature: Original
XLI: 09, 199-202, LNM 1934 (2008)
MARKOWSKY, Greg
Proof of a Tanaka-like formula stated by J. Rosen in Séminaire XXXVIII (
Stochastic calculus)
Nature: Original
XLI: 10, 203-213, LNM 1934 (2008)
BAILLEUL, Ismael
Une preuve simple d'un résultat de DufresneNature: Original
XLI: 11, 215-232, LNM 1934 (2008)
SERLET, Laurent
Creation or deletion of a drift on a Brownian trajectoryNature: Original
XLI: 12, 233-264, LNM 1934 (2008)
COX, A. M. G.
Extending Chacon-Walsh: minimality and generalised starting distributionsNature: Original
XLI: 13, 265-278, LNM 1934 (2008)
BROSSARD, Jean;
LEURIDAN, Christophe
Transformations browniennes et compléments indépendants: résultats et problèmes ouvertsNature: Original
XLI: 14, 279-294, LNM 1934 (2008)
GRUET, Jean-Claude
Hyperbolic random walksNature: Original
XLI: 15, 295-347, LNM 1934 (2008)
BAKRY, D.;
HUET, N.
The hypergroup property and representation of Markov kernelsNature: Original
XLI: 16, 349-369, LNM 1934 (2008)
WILLIAMS, David
A new look at `Markovian' Wiener-Hopf theoryNature: Original
XLI: 17, 371-377, LNM 1934 (2008)
BOLLEY, F.
Separability and completeness for the Wasserstein distanceNature: Original
XLI: 18, 379-399, LNM 1934 (2008)
PRIVAULT, Nicolas
A probabilistic interpretation to the symmetries of a discrete heat equationNature: Original
XLI: 19, 401-420, LNM 1934 (2008)
KAJI, Shunsuke
On the tail distributions of the supremum and the quadratic variation of a càdlàg local martingale (
Theory of martingales)
Nature: Original
XLI: 20, 421-438, LNM 1934 (2008)
FRIZ, Peter;
VICTOIR, Nicolas
The Burkholder-Davis-Gundy inequality for enhanced martingales (
Theory of martingales,
Integration theory)
Nature: Original
XLI: 21, 439-442, LNM 1934 (2008)
KABANOV, Yuri;
STRICKER, Christophe
On martingale selectors of cone-valued processes (
Theory of martingales)
Nature: Original
XLI: 22, 443-454, LNM 1934 (2008)
KLEIN, Irene
No asymptotic free lunch reviewed in the light of Orlicz spaces (
Mathematical finance)
Nature: Original
XLI: 23, 455-462, LNM 1934 (2008)
RÁSONYI, Miklós
New methods in the arbitrage theory of financial markets with transaction costs (
Mathematical finance)
Nature: Original
XLII: 02, 103-130, LNM 1978 (2009)
MICLO, Laurent
Monotonicity of the extremal functions for one-dimensional inequalities of logarithmic Sobolev typeNature: Original
XLII: 03, 131-136, LNM 1978 (2009)
SCHACHERMAYER, Walter;
SCHMOCK, Uwe;
TEICHMANN, Josef
Non-monotone convergence in the quadratic Wasserstein distanceNature: Original
XLII: 04, 137-145, LNM 1978 (2009)
XU, Fangjun
On the equation $\mu=S_t\mu\ast\mu_t$Nature: Original
XLII: 05, 147-151, LNM 1978 (2009)
BIANE, Philippe
Shabat polynomials and harmonic measureNature: Original
XLII: 06, 153-169, LNM 1978 (2009)
DEMNI, Nizar
Radial Dunkl processes associated with dihedral systemsNature: Original
XLII: 07, 171-185, LNM 1978 (2009)
BIANE, Philippe
Matrix valued Brownian motion and a paper by PólyaNature: Original
XLII: 08, 187-227, LNM 1978 (2009)
YANO, Kouji;
YANO, Yuko;
YOR, Marc
On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes (
Theory of Lévy processes)
Nature: Original
XLII: 09, 229-259, LNM 1978 (2009)
FITZSIMMONS, P. J.;
GETOOR, R. K.
Lévy systems and time changes (
Theory of Lévy processes)
Nature: Original
XLII: 10, 261-280, LNM 1978 (2009)
KRELL, Nathalie
Self-similar branching Markov chains (
Theory of Markov chains)
Nature: Original
XLII: 11, 281-330, LNM 1978 (2009)
HARDY, Robert;
HARRIS, Simon C.
A spine approach to branching diffusions with applications to $L^p$-convergence of martingales (
Theory of martingales,
Theory of branching processes)
Nature: Original
XLII: 12, 331-363, LNM 1978 (2009)
DEBS, Pierre
Penalisation of the standard random walk by a function of the one-sided maximum of the local time or of the duration of the excursions (
Theory of stochastic processes)
Keywords: Penalisation,
Excursions,
Local timesNature: Original
XLII: 13, 365-381, LNM 1978 (2009)
ERRAOUI, M.;
ESSAKY, E. H.
Canonical representation for Gaussian processes (
Theory of Gaussian processes)
Nature: Original
XLII: 14, 383-396, LNM 1978 (2009)
ÉMERY, Michel
Recognising whether a filtration is Brownian: a case study (
Theory of Brownian motion)
Keywords: Brownian filtrationNature: Original
XLII: 15, 397-432, LNM 1978 (2009)
DHAHRI, Ameur
Markovian properties of the spin-boson modelNature: Original
XLII: 16, 433-448, LNM 1978 (2009)
ATTAL, Stéphane;
GUILLOTIN-PLANTARD, Nadine
Statistical properties of Pauli matrices going through noisy channelsNature: Original
XLIII: 01, 3-70, LNM 2006 (2011)
PICARD, Jean
Representation formulae for the fractional Brownian motion (
Theory of processes)
Keywords: Fractional Brownian motion,
Brownian motionNature: Original,
Survey
XLIII: 02, 73-94, LNM 2006 (2011)
ARNAUDON, Marc;
COULIBALY, Koléhè Abdoulaye;
THALMAIER, Anton
Horizontal diffusion in $C^1$ path space (
Theory of processes)
Keywords: Brownian motion,
Damped parallel transport,
Horizontal diffusion,
Monge-Kantorovich problem,
Ricci curvatureNature: Original
XLIII: 03, 95-104, LNM 2006 (2011)
ROSEN, Jay
A stochastic calculus proof of the CLT for the $L^{2}$ modulus of continuity of local time (
Theory of Brownian motion)
Keywords: Central Limit Theorem,
Moduli of continuity,
Local times,
Brownian motionNature: Original
XLIII: 04, 105-126, LNM 2006 (2011)
MATSUMOTO, Ayako;
YANO, Kouji
On a zero-one law for the norm process of transient random walk (
Theory of random walks)
Keywords: Zero-one law,
Random walk,
Local times,
Jeulin's lemmaNature: Original
XLIII: 05, 127-186, LNM 2006 (2011)
LAURENT, Stéphane
On standardness and I-cosiness (
Filtrations)
Keywords: Filtrations,
CosinessNature: Original,
Exposition
XLIII: 06, 187-189, LNM 2006 (2011)
DELLACHERIE, Claude
On isomorphic probability spacesNature: Original
XLIII: 07, 191-214, LNM 2006 (2011)
RIEDLE, Markus
Cylindrical Wiener ProcessesKeywords: Cylindrical Wiener process,
Cylindrical process,
Cylindrical measure,
Stochastic integrals,
Stochastic differential equations,
Radonifying operator,
Reproducing kernel Hilbert spaceNature: Original
XLIII: 09, 221-239, LNM 2006 (2011)
MAROUBY, Matthieu
Simulation of a Local Time Fractional Stable Motion (
Theory of processes)
Keywords: Stable processes,
Self-similar processes,
Shot noise series,
Local times,
Fractional Brownian motion,
SimulationNature: Original
XLIII: 10, 241-268, LNM 2006 (2011)
BÉRARD BERGERY, Blandine;
VALLOIS, Pierre
Convergence at first and second order of some approximations of stochastic integrals (
Theory of Brownian motion,
Theory of stochastic integrals)
Keywords: Stochastic integration by regularization,
Quadratic variation,
First and second order convergence,
Stochastic Fubini's theoremNature: Original
XLIII: 11, 269-307, LNM 2006 (2011)
PAGÈS, Gilles;
SELLAMI, Afef
Convergence of multi-dimensional quantized SDE's (
Integration theory,
Theory of processes)
Keywords: Functional quantization,
Stochastic differential equations,
Stratonovich integrals,
Stationary quantizers,
Rough paths,
Itô map,
Hölder semi-norm,
$p$-variationNature: Original
XLIII: 12, 309-325, LNM 2006 (2011)
TUDOR, Ciprian A.
Asymptotic Cramér's theorem and analysis on Wiener space (
Limit theorems,
Stochastic analysis)
Keywords: Multiple stochastic integrals,
Limit theorems,
Malliavin calculus,
Stein's methodNature: Original
XLIII: 13, 327-340, LNM 2006 (2011)
LEHEC, Joseph
Moments of the Gaussian Chaos (
Stochastic analysis)
Nature: Original
XLIII: 14, 341-349, LNM 2006 (2011)
BOULEAU, Nicolas
The Lent Particle Method for Marked Point Processes (
General theory of processes,
Point processes)
Nature: Original
XLIII: 15, 351-377, LNM 2006 (2011)
BOURGADE, Paul;
NIKEGHBALI, Ashkan;
ROUAULT, Alain
Ewens measures on compact groups and hypergeometric kernels (
Theory of random matrices)
Keywords: Decomposition of Haar Measure,
Random Matrices,
Characteristic Polynomial,
Ewens sampling formula,
Correlation kernelNature: Original
XLIII: 16, 379-394, LNM 2006 (2011)
ATTAL, Stéphane;
NECHITA, Ion
Discrete approximation of the free Fock space (
Non commutative probability theory)
Keywords: Free probability,
Free Fock space,
Toy Fock space,
Limit theoremsNature: Original
XLIII: 17, 395-412, LNM 2006 (2011)
CZICHOWSKY, Christoph;
WESTRAY, Nicholas;
ZHENG, Harry
Convergence in the semimartingale topology and constrained portfolios (
Martingale theory,
Mathematical finance)
Nature: Original
XLIII: 18, 413-436, LNM 2006 (2011)
CZICHOWSKY, Christoph;
SCHWEIZER, Martin
Closedness in the semimartingale topology for spaces of stochastic integrals with constrained integrandsKeywords: Stochastic integrals,
Constrained strategies,
Semimartingale topology,
Closedness,
Predictably convex,
Projection on predictable range,
Predictable correspondence,
Optimisation under constraints,
Mathematical financeNature: Original
XLIII: 19, 437-439, LNM 2006 (2011)
BAKER, David;
YOR, Marc
On martingales with given marginals and the scaling property (
Martingale theory,
Theory of Brownian motion)
Nature: Original
XLIII: 20, 441-449, LNM 2006 (2011)
BAKER, David;
DONATI-MARTIN, Catherine;
YOR, Marc
A sequence of Albin type continuous martingales with Brownian marginals and scaling (
Martingale theory)
Keywords: Martingales,
Brownian marginalsNature: Original
XLIII: 21, 451-503, LNM 2006 (2011)
HIRSCH, Francis;
PROFETA, Christophe;
ROYNETTE, Bernard;
YOR, Marc
Constructing self-similar martingales via two Skorokhod embeddings (
Martingale theory)
Keywords: Skorokhod embeddings,
Hardy-Littlewood functions,
Convex order,
Schauder fixed point theorem,
Self-similar martingales,
Karamata's representation theoremNature: Original
XLIV: 01, 1-39, LNM 2046 (2012)
CÉNAC, Peggy;
CHAUVIN, Brigitte;
PACCAUT, Frédéric;
POUYANNE, Nicolas
Context trees, variable length Markov chains and dynamical sources (
Theory of Markov chains)
Keywords: Variable length Markov chains,
Dynamical systems of the interval,
Dirichlet series,
Occurrences of words,
Probabilistic dynamical sourcesNature: Original
XLIV: 02, 41-59, LNM 2046 (2012)
MIJATOVIĆ, Aleksandar;
NOVAK, Nika;
URUSOV, Mikhail
Martingale property of generalized stochastic exponentials (
Theory of martingales)
Keywords: Generalized stochastic exponentials,
Local martingales vs. true martingales,
One-dimensional diffusionsNature: Original
XLIV: 03, 61-74, LNM 2046 (2012)
BASSE-O'CONNOR, Andreas;
GRAVERSEN, Svend-Erik;
PEDERSEN, Jan
Some classes of proper integrals and generalized Ornstein-Uhlenbeck processes (
Theory of processes)
Keywords: Stochastic integration,
Lévy processes,
Generalized Ornstein-Uhlenbeck processesNature: Original
XLIV: 04, 75-103, LNM 2046 (2012)
QIAN, Zhongmin;
YING, Jiangang
Martingale representations for diffusion processes and backward stochastic differential equations (
Stochastic calculus)
Keywords: Backward Stochastic Differential equations,
Dirichlet forms,
Hunt processes,
Martingales,
Natural filtration,
Non-linear equationsNature: Original
XLIV: 05, , LNM 2046 (2012)
MOCHA, Markus;
WESTRAY, Nicholas
Quadratic Semimartingale BSDEs Under an Exponential Moments Condition (
Stochastic calculus)
Keywords: Quadratic Semimartingale BSDEs,
Convex Generators,
Exponential MomentsNature: Original
XLIV: 06, 141-148, LNM 2046 (2012)
MARKOWSKY, Greg
The derivative of the intersection local time of Brownian motion through Wiener chaos (
Theory of processes)
Keywords: Intersection of local time,
Wiener chaosNature: Original
XLIV: 07, 149-166, LNM 2046 (2012)
WU, Hao
On the occupation times of Brownian excursions and Brownian loops (
Theory of processe)
Keywords: Conformal invariance,
Brownian excursion measure,
Brownian loop measure,
Green's functionNature: Original
XLIV: 08, 167-190, LNM 2046 (2012)
HAJRI, Hatem
Discrete approximation to solution flows of Tanaka's SDE related to Walsh Brownian motion (
Stochastic calculus,
Limit theorems)
Keywords: Walsh's Brownian motion,
Tanaka's SDE,
Local timesNature: Original
XLIV: 09, 191-206, LNM 2046 (2012)
DEMNI, Nizar;
HMIDI, Taoufik
Spectral Distribution of the Free unitary Brownian motion: another approach (
Non commutative probability theory)
Keywords: Free unitary Brownian motion,
Spectral distributionNature: Original
XLIV: 10, 207-213, LNM 2046 (2012)
EISENBAUM, Nathalie
Another failure in the analogy between Gaussian and semicircle laws (
Non commutative probability theory)
Keywords: Gaussian law,
Semicircle law,
Free Poisson distribution,
Free probability,
Free convolution,
$R$-transform,
Dynkin isomorphismNature: Original
XLIV: 11, 215-246, LNM 2046 (2012)
LEJAY, Antoine
Global solutions to rough differential equations with unbounded vector fields (
Integration theory)
Keywords: Controlled differential equations,
Rough paths,
Euler scheme,
Global solution to differential equation,
Rough differential equationNature: Original
XLIV: 12, 247-269, LNM 2046 (2012)
MARTY, Renaud;
SØLNA, Knut
Asymptotic behavior of oscillatory fractional processes (
Theory of processes,
Limit theorems)
Keywords: Fractional processes,
Brownian motion,
Waves in random mediaNature: Original
XLIV: 13, 271-277, LNM 2046 (2012)
VUOLLE-APIALA, Juha
Time inversion property for rotation invariant self-similar diffusion processes (
Theory of processes)
Keywords: Time inversion,
Self-similar,
Bessel processes,
Diffusion processes,
Rotation invariant,
Skew product,
Radial processNature: Original
XLIV: 15, 281-315, LNM 2046 (2012)
BOGSO, Antoine-Marie;
PROFETA, Christophe;
ROYNETTE, Bernard
Some examples of peacocks in a Markovian set-up (
Theory of processes)
Keywords: Processes increasing in the convex order: peacocks,
Conditionally monotone processes,
Stochastic order,
Markov processNature: Original
XLIV: 16, 317-374, LNM 2046 (2012)
BOGSO, Antoine-Marie;
PROFETA, Christophe;
ROYNETTE, Bernard
Peacocks obtained by normalisation; strong and very strong peacocks (
Theory of processes)
Keywords: Peacocks,
Conditionally monotone processes,
Strong peacocks,
Very strong peacocksNature: Original
XLIV: 17, 375-399, LNM 2046 (2012)
HARRIS, Simon C.;
ROBERTS, Matthew I.
Branching Brownian motion: Almost sure growth along scaled paths (
Limit theorems,
Theory of processes)
Keywords: Branching Brownian motionNature: Original
XLIV: 18, 401-407, LNM 2046 (2012)
MOURRAT, Jean-Christophe
On the delocalized phase of the random pinning model (
Statistical mechanics)
Keywords: Directed Polymer models,
Partition functionNature: Original
XLIV: 19, 409-428, LNM 2046 (2012)
BERCU, Bernard;
BONY, Jean-François;
BRUNEAU, Vincent
Large deviations for Gaussian stationary processes and semi-classical analysis (
Limit theorems,
theory of processes)
Keywords: Large deviations,
Gaussian processes,
Toeplitz matrices,
Distribution of eigenvaluesNature: Original
XLIV: 20, 429-465, LNM 2046 (2012)
LÉONARD, Christian
Girsanov theory under a finite entropy condition (
Theory of processes)
Keywords: Stochastic processes,
Relative entropy,
Girsanov's theory,
Diffusion processes,
Processes with jumpsNature: Original
XLV: 01, 3-89, LNM 2078 (2013)
NOURDIN, Ivan
Lectures on Gaussian Approximations with Malliavin Calculus (
Limit Theorems)
Nature: Original
XLV: 02, 93-121, LNM 2078 (2013)
PROKAJ, Vilmos
Some Sufficient Conditions for the Ergodicity of the Lévy Transformation (
Theory of processes)
Keywords: Ergodic theory,
Lévy processNature: Original
XLV: 03, 123-139, LNM 2078 (2013)
LAURENT, Stéphane
Vershik's Intermediate Level Standardness Criterion and the Scale of an AutomorphismKeywords: Filtration,
Vershik standardness criterionNature: Original
XLV: 04, 141-157, LNM 2078 (2013)
DELLACHERIE, Claude;
ÉMERY, Michel
Filtrations Indexed by Ordinals; Application to a Conjecture of S. LaurentKeywords: FiltrationNature: Original
XLV: 05, 159-165, LNM 2078 (2013)
ÉMERY, Michel
A Planar Borel Set Which Divides Every Non-negligible Borel ProductKeywords: FiltrationNature: Original
XLV: 06, 167-180, LNM 2078 (2013)
BROSSARD, Jean;
LEURIDAN, Christophe
Characterising Ocone Local Martingales with Reflections (
Theory of processes)
Keywords: Ocone matingales,
reflection principleNature: Original
XLV: 07, 181-199, LNM 2078 (2013)
HASHIMOTO, Hiroya
Approximation and Stability of Solutions of SDEs Driven by a Symmetric $\alpha$-Stable Process with Non-Lipschitz Coefficients (
Theory of processes)
Keywords: $\alpha$-stable processes,
Euler-Maruyama approximation,
stability of solutionNature: Original
XLV: 08, 201-244, LNM 2078 (2013)
CUCHIERO, Christa;
TEICHMANN, Josef
Path Properties and Regularity of Affine Processes on General State Spaces (
Theory of processes)
Keywords: affine processes,
path properties,
regularity,
Markov semimartingalesNature: Original
XLV: 09, 245-275, LNM 2078 (2013)
JACOB, Emmanuel
Langevin Process Reflected on a Partially Elastic Boundary II (
Theory of processes)
Keywords: Langevin process,
second order reflection,
recurrent extension,
excursion measure,
stochastic differential equations,
$h$-transformNature: Original
XLV: 10, 277-300, LNM 2078 (2013)
DONEY, R. A.;
VAKEROUDIS, S.
Windings of Planar Stable Processes (
Theory of processes)
Keywords: Stable processes,
Lévy processes,
Brownian motion,
windings,
exit time from a cone,
Spitzer's Theorem,
skew-product representation,
Lamperti's relation,
Law of the Iterated Logarithm for small timesNature: Original
XLV: 11, 301-304, LNM 2078 (2013)
SOKOL, Alexander
An Elementary Proof that the First Hitting Time of an Open Set by a Jump Process is a Stopping Time (
Theory of processes)
Keywords: Stopping time,
Jump process,
First hitting timeNature: Original
XLV: 12, 305-322, LNM 2078 (2013)
DÖRING, Leif;
ROBERTS, Matthew I.
Catalytic Branching Processes via Spine Techniques and Renewal TheoryKeywords: Branching processes,
renewal theoremNature: Original
XLV: 13, 323-351, LNM 2078 (2013)
BOURGUIN, Solesne;
TUDOR, Ciprian A.
Malliavin Calculus and Self Normalized Sums (
Theory of processes)
Keywords: Malliavin calculus,
Stein's method,
self-normalized sums,
limit theorems,
multiple stochastic integrals,
chaos expansionsNature: Original
XLV: 14, 353-364, LNM 2078 (2013)
CATUOGNO, Pedro J.;
LEDESMA, Diego S.;
RUFFINO, Paulo R.
A Note on Stochastic Calculus in Vector Bundles (
Theory of processes)
Keywords: Vector bundles,
global analysis,
stochastic calculusNature: Original
XLV: 15, 365-400, LNM 2078 (2013)
PAGÈS, Gilles
Functional Co-monotony of Processes with Applications to Peacocks and Barrier Options (
Theory of processes)
Keywords: Co-monotony,
antithetic simulation method,
processes with independent increments,
Liouville processes,
fractional Brownian motion,
Asian options,
sensitivity,
barrier optionsNature: Original
XLV: 16, 401-431, LNM 2078 (2013)
NOREDDINE, Salim
Fluctuations of the Traces of Complex-Valued Random Matrices (
Non commutative probability theory)
Keywords: Random matrices,
Central limit theoremNature: Original
XLV: 17, 433-458, LNM 2078 (2013)
ORTMANN, Janosch
Functionals of the Brownian Bridge (
Non commutative probability theory)
Keywords: free Brownian bridge,
semicircular random variablesNature: Original
XLV: 18, 459-481, LNM 2078 (2013)
MICLO, Laurent;
MONMARCHÉ, Pierre
Étude spectrale minutieuse de processus moins indécis que les autres (
Theory of processes)
Keywords: Non-reversible Markov processes,convergence to equilibriumNature: Original
XLV: 19, 483-535, LNM 2078 (2013)
BARTHE, Franck;
BORDENAVE, Charles
Combinatorial Optimization Over Two Random Point SetsKeywords: combinatorial optimization,
minimal matching,
geometric probabilityNature: Original
XLV: 20, 537-558, LNM 2078 (2013)
KORTCHEMSKI, Igor
A Simple Proof of Duquesne's Theorem on Contour Processes of Conditioned Galton-Watson TreesKeywords: Conditioned Galton-Watson tree,
Stable continuous random tree,
Scaling limit,
Invariance principleNature: Original
XLVI: 01, 1-32, LNM 2123 (2014)
BOCHAROV, Sergey;
HARRIS, Simon C.
Branching random walk in an homogeneous breeding potential (
Theory of branching processes)
This articles studies the explosion and non-explosion of the population described by a branching random walk, as well as the behavior of the rightmost particle.
Keywords: Branching random walk,
explosionNature: Original
XLVI: 02, 33-59, LNM 2123 (2014)
KYPRIANOU, A.E.;
PÉREZ, J.-L.;
REN, Y.X.
The backbone decomposition for spatially dependent supercritical superprocesses (
Theory of branching processes)
Keywords: SuperprocessesNature: Original
XLVI: 03, 61-70, LNM 2123 (2014)
BEZNEA, Lucian;
CÎMPEAN, Iulian
On Bochner-Kolmogorov theorem (
Theory of branching processes)
Keywords: Bochner-Kolmogorov theorem,
space of finite configurations,
space of fragmentation sizesNature: Original
XLVI: 04, 71-103, LNM 2123 (2014)
FRANCHI, Jacques
Small Time Asymptotics for an Example of Strictly Hypoelliptic Heat Kernel (
Stochastic analysis)
Nature: Original
XLVI: 05, 105-123, LNM 2123 (2014)
COULIBALY-PASQUIER, ;
Koléhè, A.
Onsager-Machlup functional for uniformly elliptic time-inhomogeneous diffusion (
Stochastic analysis)
Nature: Original
XLVI: 06, 125-193, LNM 2123 (2014)
GENG, Xi;
QIAN, Zhongmin;
YANG, Danyu
$G$-Brownian Motion as Rough Paths and Differential Equations Driven by $G$-Brownian Motion (
Stochastic analysis)
This article studies stochastic differential equations driven by the $G$-Brownian motion in the context of rough paths theory
Keywords: rough path,
$G$-Brownian motionNature: Original
XLVI: 10, 195-205, LNM 2123 (2014)
BAILLEUL, Ismaël
Flows driven by Banach space-valued rough paths (
Miscellanea)
This article gives a quick proof of the existence of a solution to a rough differential equations by constructing directly its associated flow. Bound on the solutions, which are valid in an infinite dimensional setting, are also given
Keywords: rough pathsNature: Original
XLVI: 07, 207-230, LNM 2123 (2014)
LÉONARD, Christian
Some properties of path measures (
Theory of processes)
Keywords: unbounded measure,
conditional expectation,
relative entropy,
stochastic processes,
Schrödinger problemNature: Original
XLVI: 08, 231-292, LNM 2123 (2014)
CATTIAUX, Patrick;
GUILLIN, Arnaud
Semi Log-Concave Markov Diffusions (
Theory of processes)
Nature: Original
XLVI: 09, 293-315, LNM 2123 (2014)
MARINELLI, Carlo;
RÖCKNER, Michael
On maximal inequalities for purely discontinuous martingales in infinite dimensionsNature: Original
XLVI: 11, 317-331, LNM 2123 (2014)
SCHACHERMAYER, Walter
Admissible Trading Strategies under Transaction CostsNature: Original
XLVI: 12, 333-343, LNM 2123 (2014)
KYPRIANOU, A.E.;
WATSON, A.R.
Potentials of stable processesNature: Original
XLVI: 13, 345-357, LNM 2123 (2014)
LETEMPLIER, Julien;
SIMON, Thomas
Unimodality of hitting times for stable processes (
Theory of stable processes)
This article shows that the hitting times of points of a real $\alpha$-stable Lévy process are unimodal random variables
Keywords: $\alpha$-stable Lévy process,
self-decomposability,
Kanter random variable,
size-biasNature: Original
XLVI: 14, 359-375, LNM 2123 (2014)
ROSENBAUM, Mathieu;
YOR, Marc
On the law of a triplet associated with the pseudo-Brownian bridge (
Theory of Brownian motion)
This article gives a remarkable identity in law which relates the Brownian motion, its local time, and the the inverse of its local time
Keywords: Brownian motion,
pseudo-Brownian bridge,
Bessel process,
local time,
hitting times,
scaling,
uniform sampling,
Mellin transformNature: Original
XLVI: 15, 377-394, LNM 2123 (2014)
BROSSARD, Jean;
ÉMERY, Michel;
LEURIDAN, Christophe
Skew-product decomposition of planar Brownian motion and complementabilityNature: Original
XLVI: 16, 395-400, LNM 2123 (2014)
PROKAJ, Vilmos
On the exactness of the Lévy-transformationNature: Original
XLVI: 17, 401-410, LNM 2123 (2014)
CHANG, Yinshan
Multi-occupation field generates the Borel-sigma-field of loopsNature: Original
XLVI: 18, 411-459, LNM 2123 (2014)
VAN HANDEL, Ramon
Ergodicity, Decisions, and Partial InformationNature: Original
XLVI: 19, 461-472, LNM 2123 (2014)
SERLET, Laurent
Invariance principle for the random walk conditioned to have a few zeroesNature: Original
XLVI: 21, 481-512, LNM 2123 (2014)
NAJNUDEL, Joseph;
NIKEGHBALI, Ashkan
On a flow of operators associated to virtual permutationsNature: Original
XLVII: 03, 1-15, LNM 2137 (2015)
SALMINEN, Paavo;
YEN, Ju-Yi;
YOR, Marc
Integral Representations of Certain Measures in the One-Dimensional Diffusions Excursion TheoryNature: Original
XLVII: 04, 17-35, LNM 2137 (2015)
WARREN, Jon
Sticky Particles and Stochastic FlowsNature: Original
XLVII: 05, 37-47, LNM 2137 (2015)
FUNAKI, Tadahisa
Infinitesimal Invariance for the Coupled KPZ EquationsNature: Original
XLVII: 06, 49-88, LNM 2137 (2015)
PITMAN, Jim;
TANG, Wenpin
Patterns in Random Walks and Brownian MotionNature: Original
XLVII: 07, 89-105, LNM 2137 (2015)
LE GALL, Jean-François
Bessel Processes, the Brownian Snake and Super-Brownian MotionNature: Original
XLVII: 08, 107-126, LNM 2137 (2015)
ALILI, Larbi;
GRACZYK, Piotr;
.{Z}AK, Tomasz
On Inversions and Doob $h$-Transforms of Linear DiffusionsNature: Original
XLVII: 09, 127-156, LNM 2137 (2015)
YANO, Kouji;
YANO, Yuko
On $h$-Transforms of One-Dimensional Diffusions Stopped upon Hitting ZeroNature: Original
XLVII: 10, 157-185, LNM 2137 (2015)
BAKRY, Dominique;
ZRIBI, Olfa
$h$-Transforms and Orthogonal PolynomialsNature: Original
XLVII: 11, 187-218, LNM 2137 (2015)
AKSAMIT, Anna;
CHOULLI, Tahir;
JEANBLANC, Monique
On an Optional Semimartingale Decomposition and the Existence of a Deflator in an Enlarged FiltrationNature: Original
XLVII: 12, 219-225, LNM 2137 (2015)
PITMAN, Jim
Martingale Marginals Do Not Always Determine ConvergenceNature: Original
XLVII: 13, 227-247, LNM 2137 (2015)
OB\L{}ÓJ, Jan;
SPOIDA, Peter;
TOUZI, Nizar
Martingale Inequalities for the Maximum via Pathwise ArgumentsNature: Original
XLVII: 14, 249-262, LNM 2137 (2015)
BIANE, Philippe
Polynomials Associated with Finite Markov ChainsNature: Original
XLVII: 15, 263-297, LNM 2137 (2015)
NAJNUDEL, Joseph
On $\sigma$-Finite Measures Related to the Martin Boundary of Recurrent Markov ChainsNature: Original
XLVII: 16, 299-320, LNM 2137 (2015)
FITZSIMMONS, Pat;
LE JAN, Yves;
ROSEN, Jay
Loop Measures Without Transition ProbabilitiesNature: Original
XLVII: 17, 321-338, LNM 2137 (2015)
DUEMBGEN, Moritz;
ROGERS, L. C. G.
The Joint Law of the Extrema, Final Value and Signature of a Stopped Random WalkNature: Original
XLVII: 18, 339-367, LNM 2137 (2015)
AZMOODEH, Ehsan;
PECCATI, Giovanni;
POLY, Guillaume
Convergence Towards Linear Combinations of Chi-Squared Random Variables: A Malliavin-Based ApproachNature: Original
XLVII: 19, 369-425, LNM 2137 (2015)
MÉLIOT, Pierre-Loïc;
NIKEGHBALI, Ashkan
Mod-Gaussian Convergence and Its Applications for Models of Statistical MechanicsNature: Original
XLVII: 20, 427-441, LNM 2137 (2015)
BALDI, Paolo;
CARAMELLINO, Lucia;
ROSSI, Maurizia
On Sharp Large Deviations for the Bridge of a General DiffusionNature: Original
XLVII: 21, 443-466, LNM 2137 (2015)
DEMNI, Nizar;
ROUAULT, Alain;
ZANI, Marguerite
Large Deviations for Clocks of Self-similar ProcessesNature: Original
XLVII: 22, 467-496, LNM 2137 (2015)
O'CONNELL, Neil
Stochastic Bäcklund TransformationsNature: Original
XLVII: 23, 497-504, LNM 2137 (2015)
IKEDA, Nobuyuki;
MATSUMOTO, Hiroyuki
The Kolmogorov Operator and Classical MechanicsNature: Original
XLVII: 24, 505-519, LNM 2137 (2015)
COMTET, Alain;
TOURIGNY, Yves
Explicit Formulae in Probability and in Statistical PhysicsNature: Original
XLVII: 25, 521-559, LNM 2137 (2015)
BOUGEROL, Philippe
Matsumoto-Yor Process and Infinite Dimensional Hyperbolic SpaceNature: Original
XLVII: 26, 561-584, LNM 2137 (2015)
CHAUMONT, Loïc
Breadth First Search Coding of Multitype Forests with Application to RepresentationNature: Original
XLVII: 27, 585-601, LNM 2137 (2015)
DEVROYE, Luc;
LETAC, Gérard
Copulas with Prescribed Correlation MatrixNature: Original
XLVII: 28, 603-617, LNM 2137 (2015)
STROOCK, W. Daniel
Remarks on the HRT ConjectureNature: Original
