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: 23, 253-259, LNM 721 (1979)
SPILIOTIS, Jean
Sur les intégrales stochastiques de L.C. Young (
Stochastic calculus)
This is a partial exposition of a theory of stochastic integration due to L.C. Young (
Advances in Prob. 3, 1974)
Keywords: Stochastic integralsNature: Exposition 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: 45, 521-532, LNM 721 (1979)
JEULIN, Thierry
Un théorème de J.W. Pitman (
Brownian motion,
Diffusion theory)
This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$
Keywords: Bessel processesNature: New proof of known results 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: 52, 611-613, LNM 721 (1979)
MEYER, Paul-André
Présentation de l'``inégalité de Doob'' de Métivier et Pellaumail (
Martingale theory)
In the theory of stochastic differential equations with respect to discontinuous semimartingales, stopping processes ``just before'' a stopping time $T$ (at $T-$) is a basic technique, but since it does not preserve the martingale property, Doob's inequality cannot be used to control the stopped process. The inequality discussed here is an efficient substitute, used by Métivier-Pellaumail (
Ann. Prob. 8, 1980) to develop the whole theory of stochastic differential equations
Keywords: Doob's inequality,
Stochastic differential equationsNature: Exposition 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: 54, 620-623, LNM 721 (1979)
MEYER, Paul-André
Caractérisation des semimartingales, d'après Dellacherie (
Stochastic calculus)
This short paper contains the proof of a very important theorem, due to Dellacherie (with the crucial help of Mokobodzki for the functional analytic part). Namely, semimartingales are exactly the processes which give rise to a nice vector measure on the previsible $\sigma$-field, with values in the (non locally convex) space $L^0$. It is only fair to say that this direction was initiated by Métivier and Pellaumail, and that the main result was independently discovered by Bichteler,
Ann. Prob. 9, 1981)
Comment: An important lemma which simplifies the proof and has other applications is given by Yan in
1425Keywords: Semimartingales,
Stochastic integralsNature: Exposition Retrieve article from Numdam
XIII: 55, 624-624, LNM 721 (1979)
YOR, Marc
Un exemple de J. Pitman (
General theory of processes)
The balayage formula allows the construction of many martingales vanishing on the zeros of a given continuous martingale $X$, namely martingales of the form $Z_{g_t}X_t$ where $Z$ is previsible. Taking $X$ to be Brownian motion, an example is given of a martingale vanishing on its zeros which is not of the above form
Comment: The general problem of finding all martingales which vanish on the zeros of a given continuous martingale is discussed by Azéma and Yor in
2622Keywords: Balayage,
Balayage formulaNature: Exposition 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
XIII: 60, 647-647, LNM 721 (1979)
BRETAGNOLLE, Jean;
HUBER, Catherine
Corrections à un exposé antérieur (
Mathematical statistics)
Two misprints and a more substantial error (in the proof of proposition 1) of
1224 are corrected
Comment: A revised version appeared in (
Zeit. für W-Theorie, 47, 1979)
Keywords: Empirical distribution function,
Prohorov distanceNature: Correction Retrieve article from Numdam
