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
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
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
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
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: 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: 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
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: 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
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: 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: 51, 740-740, LNM 649 (1978)
MEYER, Paul-André
Correction à ``Sur un théorème de C. Stricker'' (
Stochastic calculus)
Fills a gap in a proof in
1132Keywords: Stochastic integralsNature: Correction 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: 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: 57, 763-769, LNM 649 (1978)
MEYER, Paul-André
La formule d'Ito pour le mouvement brownien, d'après Brosamler (
Brownian motion,
Stochastic calculus)
This paper presents the results of a paper by Brosamler (
Trans. Amer. Math. Soc. 149, 1970) on the Ito formula $f(B_t)=...$ for $n$-dimensional Brownian motion, under the weakest possible assumptions: namely up to the first exit time from an open set $W$ and assuming only that $f$ is locally in $L^1$ in $W$, and its Laplacian in the sense of distributions is a measure in $W$
Keywords: Ito formulaNature: Exposition 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: 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: 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: 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
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: 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: 14, 125-127, LNM 784 (1980)
LENGLART, Érik
Sur l'inégalité de Métivier-Pellaumail (
Stochastic calculus)
A simplified (but still not so simple) proof of the Métivier-Pellaumail inequality
Keywords: Doob's inequality,
Métivier-Pellaumail inequalityNature: New proof of known results 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: 20, 173-188, LNM 784 (1980)
MEYER, Paul-André
Les résultats de Jeulin sur le grossissement des tribus (
General theory of processes,
Stochastic calculus)
This is an introduction to beautiful results of Jeulin on enlargements, for which see
Zeit. für W-Theorie, 52, 1980, and above all the Lecture Notes vol. 833,
Semimartingales et grossissement d'une filtration Comment: See also
1329,
1350Keywords: Enlargement of filtrations,
SemimartingalesNature: Exposition Retrieve article from Numdam
XIV: 24, 209-219, LNM 784 (1980)
PELLAUMAIL, Jean
Remarques sur l'intégrale stochastique (
Stochastic calculus)
This is an exposition of stochastic integrals and stochastic differential equations for Banach space valued processes along the lines of Métivier-Pellaumail
Stochastic Integration (1980), the class of semimartingales being defined by the Métivier-Pellaumail inequality (
1413)
Keywords: Stochastic integrals,
Stochastic differential equations,
Métivier-Pellaumail inequalityNature: Exposition 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: 29, 254-254, LNM 784 (1980)
YOEURP, Chantha
Rectificatif à l'exposé de C.S. Chou (
Stochastic calculus)
A mistake in the proof of
1337 is corrected, the result remaining true without additional assumptions
Keywords: Local times,
Semimartingales,
JumpsNature: Correction 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: 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: 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: 08, 142-142, LNM 850 (1981)
MEYER, Paul-André
Une question de théorie des processus (
Stochastic calculus)
It is remarked that the stochastic integrals that appear in stochastic differential geometry are of a particular kind, and asked whether the theory could be developed for processes belonging to a larger class than semimartingales
Comment: For recent work in this area, see T. Lyons' article in
Rev. Math. Iberoamericana 14 (1998) on differential equations driven by non-smooth functions
Keywords: SemimartingalesNature: Open question 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: 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: 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: 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
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: 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: 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: 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: 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
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
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
XX: 02, 28-29, LNM 1204 (1986)
FAGNOLA, Franco;
LETTA, Giorgio
Sur la représentation intégrale des martingales du processus de Poisson (
Stochastic calculus,
Point processes)
Dellacherie gave in
805 a proof by stochastic calculus of the previsible representation property for the Wiener and Poisson processes. A gap in this proof is filled in
928 for Brownian motion and here for Poisson processes
Keywords: Stochastic integrals,
Previsible representation,
Poisson processesNature: Correction 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
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
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
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
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: 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
