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: 03, 27-33, LNM 581 (1977)
CHUNG, Kai Lai
Pedagogic notes on the barrier theorem (
Potential theory)
Let $D$ a bounded open set in $
R^n$, and let $z$ be a boundary point. Then a barrier at $z$ is a superharmonic function in $D$, strictly positive and with a limit equal to $0$ at $z$. The barrier theorem asserts that if there is a barrier at $z$, then $z$ is regular. An elegant proof of this is given using Brownian motion. Then it is shown that the expectation of $S$, the hitting time of $D^c$, is bounded, upper semi-continuous in $R^n$ and continuous in $D$, and is a barrier at every regular point
Comment: An error is corrected in
1247Keywords: Classical potential theory,
Barrier,
Regular pointsNature: New proof of known results 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: 10, 109-119, LNM 581 (1977)
MEYER, Paul-André
Convergence faible de processus, d'après Mokobodzki (
General theory of processes)
The following simple question of Benveniste was answered positively by Mokobodzki (
Séminaire de Théorie du Potentiel, Lect. Notes in M. 563, 1976): given a sequence $(U^n)$ of optional processes such that $U^n_T$ converges weakly in $L^1$ for every stopping time $T$, does there exist an optional process $U$ such that $U^n_T$ converges to $U_T$? The proof is rather elaborate
Keywords: Weak convergence in $L^1$Nature: Exposition Retrieve article from Numdam
XI: 11, 120-131, LNM 581 (1977)
MEYER, Paul-André
Résultats récents de A. Benveniste en théorie des flots (
Ergodic theory)
A filtered flow is said to be diffuse if there exists a r.v. ${\cal F}_0$-measurable $J$ such that given any ${\cal F}_0$-measurable r.v.'s $T$ and $H$, $P\{J\circ\theta_T=H, 0<T<\infty\}=0$. The main result of the paper is the fact that a diffuse flow contains all Lévy flows (flows of increments of Lévy processes, no invariant measure is involved). In particular, the Brownian flow contains a Poisson counter
Comment: This result on the whole line is similar to
1106, which concerns a half-line. The original paper of Benveniste appeared in
Z. für W-theorie, 41, 1977/78
Keywords: Filtered flows,
Poisson flowNature: Exposition 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
XI: 38, 539-565, LNM 581 (1977)
TORTRAT, Albert
Désintégration d'une probabilité. Statistiques exhaustives (
Measure theory)
This is a detailed review (including proofs) of the problem of existence of conditional distributions
Keywords: Disintegration of measures,
Conditional distributions,
Sufficient statisticsNature: Exposition Retrieve article from Numdam
XI: 39, 566-573, LNM 581 (1977)
ÉMERY, Michel
Information associée à un semigroupe (
Markov processes)
This paper contains the proof of two important theorems of Donsker and Varadhan (
Comm. Pure and Appl. Math., 1975)
Nature: Exposition Retrieve article from Numdam
