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
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: 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
XII: 03, 22-34, LNM 649 (1978)
JACOD, Jean
Projection prévisible et décomposition multiplicative d'une semi-martingale positive (
General theory of processes)
The problem discussed is the decomposition of a positive ($\ge0$) special semimartingale $X$ (the most interesting cases being super- and submartingales) into a product of a positive local martingale and a positive previsible process of finite variation. The problem is solved here in the greatest possible generality, on a maximal non-vanishing domain for $X$---this is a previsible stochastic interval $[0,S)$ which at $S$ may be open or closed
Comment: This papers improves on
1021 and
1023Keywords: Semimartingales,
Multiplicative decompositionNature: Original Retrieve article from Numdam
XII: 04, 35-46, LNM 649 (1978)
MÉMIN, Jean
Décompositions multiplicatives de semimartingales exponentielles et applications (
General theory of processes)
It is shown that, given two semimartingales $U,V$ such that $U$ has no jump equal to $-1$, there is a unique semimartingale $X$ such that ${\cal E}(X)\,{\cal E}(U)={\cal E}(V)$. This result is applied to recover all known results on multiplicative decompositions
Comment: The results of this paper are used in Mémin-Shiryaev
1312Keywords: Stochastic exponentials,
Semimartingales,
Multiplicative decompositionNature: Original Retrieve article from Numdam
XII: 08, 57-60, LNM 649 (1978)
MEYER, Paul-André
Sur un théorème de J. Jacod (
General theory of processes)
Consider a given process $X$ adapted to a given filtration $({\cal F}_t)$. The set of laws of semimartingales consists of those laws $P$ under which $X$ is a semimartingale with respect to $({\cal F}_t)$ suitably completed. Jacod proved that the set of laws of semimartingales is convex. This is extended here to countable convex combinations, and to integrals
Comment: This easy paper has some historical interest, as it raised the problem of initial enlargement of a filtration
Keywords: Semimartingales,
Enlargement of filtrations,
Laws of semimartingalesNature: Original Retrieve article from Numdam
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: 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: 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: 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: 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: 08, 76-101, LNM 784 (1980)
SHARPE, Michael J.
Local times and singularities of continuous local martingales (
Martingale theory)
This paper studies continuous local martingales $(M_t)$ in the open interval $]0,\infty[$. After recalling a few useful results on local martingales, the author proves that the sample paths a.s., either have a limit (possibly $\pm\infty$) at $t=0$, or oscillate over the whole interval $]-\infty,\infty[$ (this is due to Walsh
1133, but the proof here does not use conformal martingales). Then the quadratic variation and local time of $M$ are defined as random measures which may explode near $0$, and it is shown that non-explosion of the quadratic variation (of the local time) measure characterizes the sample paths which have a finite limit (a limit) at $0$. The results are extended in part to local martingale increment processes, which are shown to be stochastic integrals with respect to true local martingales, of previsible processes which are not integrable near $0$
Comment: See Calais-Genin
1717Keywords: Local times,
Local martingales,
Semimartingales in an open intervalNature: Original Retrieve article from Numdam
XIV: 09, 102-103, LNM 784 (1980)
MEYER, Paul-André
Sur un résultat de L. Schwartz (
Martingale theory)
the following definition of a semimartingale $X$ in a random open set $A$ is due to L. Schwartz (
Semimartingales dans les variétés..., Lecture Notes in M.
780): $A$ can be represented as a countable union of random open sets $A_n$, and for each $n$ there exists an ordinary semimartingale $Y_n$ such $X=Y_n$ on $A_n$. It is shown that if $K\subset A$ is a compact optional set, then there exists an ordinary semimartingale $Y$ such that $X=Y$ on $K$
Comment: The results are extended in Meyer-Stricker
Stochastic Analysis and Applications, part B, Advances in M. Supplementary Studies, 1981
Keywords: Semimartingales in a random open setNature: Exposition,
Original additions Retrieve article from Numdam
XIV: 10, 104-111, LNM 784 (1980)
STRICKER, Christophe
Prolongement des semi-martingales (
Stochastic calculus)
The problem consists in characterizing semimartingales on $]0,\infty[$ which can be ``closed at infinity'', and the similar problem at $0$. The criteria are similar to the Vitali-Hahn-Saks theorem and involve convergence in probability of suitable stochastic integrals. The proof rests on a functional analytic result of Maurey-Pisier
Keywords: Semimartingales,
Semimartingales in an open intervalNature: Original Retrieve article from Numdam
XIV: 11, 112-115, LNM 784 (1980)
STRICKER, Christophe
Projection optionnelle des semi-martingales (
Stochastic calculus)
Let $({\cal G}_t)$ be a subfiltration of $({\cal F}_t)$. Since the optional projection on $({\cal G}_t)$ of a ${\cal F}$-martingale is a ${\cal G}$-martingale, and the projection of an increasing process a ${\cal G}$-submartingale, projections of ${\cal F}$-semimartingales ``should be'' ${\cal G}$-semimartingales. This is true for quasimartingales, but false in general
Comment: The main results on subfiltrations are proved by Stricker in
Zeit. für W-Theorie, 39, 1977
Keywords: Semimartingales,
Projection theoremsNature: Original Retrieve article from Numdam
XIV: 12, 116-117, LNM 784 (1980)
CHOU, Ching Sung
Une caractérisation des semimartingales spéciales (
Stochastic calculus)
This is a useful addition to the next paper
1413: a semimartingale can be ``controlled'' (in the sense of Métivier-Pellaumail) by a locally integrable increasing process if and only if it is special
Comment: See also
1352Keywords: Semimartingales,
Métivier-Pellaumail inequality,
Special semimartingalesNature: Original Retrieve article from Numdam
XIV: 13, 118-124, LNM 784 (1980)
ÉMERY, Michel
Équations différentielles stochastiques. La méthode de Métivier-Pellaumail (
Stochastic calculus)
Métivier-Pellaumail introduced the idea of an increasing process $(A_t)$ controlling a semimartingale $X$ as the property $$E[\,(sup_{t<T} \int_0^t H_s dX_s)^2\,] \le E[\,A_{T-}\,\int_0^{T-} H_s^2 dA_s\,]$$ for all stopping times $T$ and bounded previsible processes $(H_t)$. For a proof see
1414. Métivier-Pellaumail used this inequality to develop the theory of stochastic differential equations (including stability) without localization and pasting together at jump times. Here their method is applied to the topology of semimartingales
Comment: See
1352. A general reference on the Métivier-Pellaumail method can be found in their book
Stochastic Integration, Academic Press 1980. See also He-Wang-Yan,
Semimartingale Theory and Stochastic Calculus, CRC Press 1992
Keywords: Semimartingales,
Spaces of semimartingales,
Stochastic differential equations,
Doob's inequality,
Métivier-Pellaumail inequalityNature: Original Retrieve article from Numdam
XIV: 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: 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: 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: 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
XV: 05, 44-102, LNM 850 (1981)
MEYER, Paul-André
Géométrie stochastique sans larmes (
Stochastic differential geometry)
Brownian motion in manifolds has been studied for many years; Ito had very early defined parallel transport along random paths, and Dynkin had extended it to tensors; Malliavin had introduced many geometric ideas into the theory of stochastic differential equations, and interest had been aroused by the ``Malliavin Calculus'' in the early eighties. The main topic of the present paper (or rather exposition: the paper contains definitions, explanations, but practically no theorems) is
continuous semimartingales in manifolds, following L.~Schwartz (LN
780, 1980), but with additional features: an indication of J.M.~Bismut hinting to a definition of continuous
martingales in a manifold, and the author's own interest on the forgotten intrinsic definition of the second differential $d^2f$ of a function. All this fits together into a geometric approach to semimartingales, and a probabilistic approach to such geometric topics as torsion-free connexions
Comment: A short introduction by the same author can be found in
Stochastic Integrals, Springer LNM 851. The same ideas are expanded and presented in the supplement to Volume XVI and the book by Émery,
Stochastic Calculus on Manifolds Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Transfer principle,
Stochastic differential equations,
Stochastic integrals,
Stratonovich integralsNature: Original Retrieve article from Numdam
XV: 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: 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: 47, 671-672, LNM 850 (1981)
BAKRY, Dominique
Une remarque sur les semi-martingales à deux indices (
Several parameter processes)
Let $({\cal F}^1_s)$ and $({\cal F}^2_t)$ be two filtrations whose conditional expectations commute. Let $(A_t)$ be a bounded increasing process adapted to $({\cal F}^2_t)$. It had been proved under stringent absolute continuity conditions on $A$ that the process $X_{st}=E[A_t\,|\,{\cal F}^1_s]$ was a semimartingale (a stochastic integrator). A counterexample is given here to show that this is not true in general
Keywords: Two-parameter semimartingalesNature: Original Retrieve article from Numdam
XVI: 15, 209-211, LNM 920 (1982)
BARLOW, Martin T.
$L(B_t,t)$ is not a semimartingale (
Brownian motion)
The following question had been open for some time: given a jointly continuous version $L(a,t)$ of the local times of Brownian motion, is $Y_t=L(B_t,t)$ a semimartingale? It is proved here that $Y$ fails to be Hölder continuous of order 1/4, and therefore cannot be a semimartingale
Keywords: Local times,
SemimartingalesNature: Original Retrieve article from Numdam
XVI: 16, 212-212, LNM 920 (1982)
WALSH, John B.
A non-reversible semi-martingale (
Stochastic calculus)
A simple example is given of a continuous semimartingale (a Brownian motion which stops at time $1$ and starts moving again at time $T>1$, $T$ encoding all the information up to time $1$) whose reversed process is not a semimartingale
Keywords: Semimartingales,
Time reversalNature: Original Retrieve article from Numdam
XVI: 17, 213-218, LNM 920 (1982)
FALKNER, Neil;
STRICKER, Christophe;
YOR, Marc
Temps d'arrêt riches et applications (
General theory of processes)
This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see
1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper
1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$
Keywords: Stopping times,
Local times,
Semimartingales,
Previsible processesNature: Original Retrieve article from Numdam
XVI-S: 57, 165-207, LNM 921 (1982)
MEYER, Paul-André
Géométrie différentielle stochastique (bis) (
Stochastic differential geometry)
A sequel to
1505. The main theme is that an ordinary differential equation has a non unique extension as a stochastic differential equation: besides the Stratonovich one, given by the ``transfer principle'', there are other possibilities: choosing among them requires some additional, connection-like, structure. The most striking application is the Dohrn-Guerra correction to the parallel transport along a semimartingale
Comment: For complements, see Émery
1658, Hakim-Dowek-Lépingle
2023, Émery's monography
Stochastic Calculus in Manifolds (Springer, 1989) and article
2428, and Arnaudon-Thalmaier
3214Keywords: Semimartingales in manifolds,
Stochastic differential equations,
Local characteristics,
Nelson's stochastic mechanics,
Transfer principleNature: Original Retrieve article from Numdam
XVI-S: 58, 208-216, LNM 921 (1982)
ÉMERY, Michel
En marge de l'exposé de Meyer : ``Géométrie différentielle stochastique'' (
Stochastic differential geometry)
Marginal remarks to Meyer
1657Keywords: Semimartingales in manifolds,
Stochastic differential equationsNature: Original Retrieve article from Numdam
XVI-S: 59, 217-236, LNM 921 (1982)
DARLING, Richard W.R.
Martingales in manifolds - Definition, examples and behaviour under maps (
Stochastic differential geometry)
Martingales in manifolds have been introduced independently by Meyer
1505 and the author (Ph.D. Thesis). This short note is a review of that thesis; here, the definition of a manifold-valued martingale is by its behaviour under convex functions
Comment: More details are given in
Bull. L.M.S. 15 (1983),
Publ R.I.M.S. Kyoto~
19 (1983) and
Zeit. für W-theorie 65 (1984). Characterizating of manifold-valued martingales by convex functions has become a powerful tool: see for instance Émery's book
Stochastic Calculus in Manifolds (Springer, 1989) and his St-Flour lectures (Springer LNM 1738)
Keywords: Martingales in manifolds,
Semimartingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
XVII: 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
XVIII: 33, 501-518, LNM 1059 (1984)
ÉMERY, Michel;
ZHENG, Wei-An
Fonctions convexes et semimartingales dans une variété (
Stochastic differential geometry)
On a manifold endowed with a connexion, convex functions can be defined, and transform manifold-valued martingales into real-valued local submartingales (see Darling
1659). This is extended here to the case of non-smooth convex functions. Ii is also shown that they make manifold-valued semimartingales into real semimartingales
Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Convex functionsNature: Original Retrieve article from Numdam
XX: 23, 352-374, LNM 1204 (1986)
HAKIM-DOWEK, M.;
LÉPINGLE, Dominique
L'exponentielle stochastique des groupes de Lie (
Stochastic differential geometry)
Given a Lie group $G$ and its Lie algebra $\cal G$, this article defines and studies the stochastic exponential of a (continuous) semimartingale $M$ in $\cal G$ as the solution in $G$ to the Stratonovich s.d.e. $dX = X dM$. The inverse operation (stochastic logarithm) is also considered; various formulas are established (e.g. the exponential of $M+N$). When $M$ is a local martingale, $X$ is a martingale for the connection such that $\nabla_A B=0$ for all left-invariant vector fields $A$ and $B$
Comment: See also Karandikar
Ann. Prob. 10 (1982) and
1722. For a sequel, see Arnaudon
2612Keywords: Semimartingales in manifolds,
Martingales in manifolds,
Lie groupNature: Original Retrieve article from Numdam
XXIV: 28, 407-441, LNM 1426 (1990)
ÉMERY, Michel
On two transfer principles in stochastic differential geometry (
Stochastic differential geometry)
Second-order stochastic calculus gives two intrinsic methods to transform an ordinary differential equation into a stochastic one (see Meyer
1657, Schwartz
1655 or Emery
Stochastic calculus in manifolds). The first one gives a Stratonovich SDE and needs coefficients regular enough; the second one gives an Ito equation and needs a connection on the manifold. Discretizing time and smoothly interpolating the driving semimartingale is known to give an approximation to the Stratonovich transfer; it is shown here that another discretized-time procedure converges to the Ito transfer. As an application, if the ODE makes geodesics to geodesics, then the Ito and Stratonovich SDE's are the same
Comment: An error is corrected in
2649. The term ``transfer principle'' was coined by Malliavin,
Géométrie Différentielle Stochastique, Presses de l'Université de Montréal (1978); see also Bismut,
Principes de Mécanique Aléatoire (1981) and
1505Keywords: Stochastic differential equations,
Semimartingales in manifolds,
Transfer principleNature: Original Retrieve article from Numdam
XXV: 18, 196-219, LNM 1485 (1991)
PICARD, Jean
Calcul stochastique avec sauts sur une variété (
Stochastic differential geometry)
It is known from Meyer
1505 that intrinsic Ito integrals have a meaning for continuous semimartingales in a manifold $M$, provided $M$ is endowed with a connection. This is extended here to càdlàg semimartingales. The manifold must be endowed with a richer structure, a ``connector'', mapping $M\times M$ to the tangent bundle, that allows to interpret a jump $(X_{t-},X_t)$ as a tangent vector to $M$ at $X{t-}$; the differential of the connector at the diagonal reduces to a classical torsion-free connection. Introducing torsions leads to a more general ``transporter'', describing how parallel transports should behave at jump times, and reducing to a classical connection for infinitesimal jumps. Discrete-time approximations are established.
Keywords: Semimartingales in manifolds,
Martingales in manifolds,
JumpsNature: Original Retrieve article from Numdam
XXVI: 18, 189-209, LNM 1526 (1992)
NORRIS, James R.
A complete differential formalism for stochastic calculus in manifolds (
Stochastic differential geometry)
The use of equivariant coordinates in stochastic differential geometry is replaced here by an equivalent, but intrinsic, formalism, where the differential of a semimartingale lives in the tangent bundle. Simple, intrinsic Girsanov and Feynman-Kac formulas are given, as well as a nice construction of a Brownian motion in a manifold admitting a Riemannian submersion with totally geodesic fibres
Keywords: Semimartingales in manifolds,
Stochastic integrals,
Feynman-Kac formula,
Changes of measure,
Heat semigroupNature: Original Retrieve article from Numdam
XXVI: 49, 633-633, LNM 1526 (1992)
ÉMERY, Michel
Correction au Séminaire~XXIV (
Stochastic differential geometry)
An error in
2428 is pointed out; it is corrected by Cohen (
Stochastics Stochastics Rep. 56, 1996)
Keywords: Stochastic differential equations,
Semimartingales in manifoldsNature: Correction Retrieve article from Numdam
XXIX: 16, 166-180, LNM 1613 (1995)
APPLEBAUM, David
A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (
Stochastic differential geometry,
Markov processes)
This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.
Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (
Stochastics Stochastics Rep. 56, 1996). The same question is addressed by Cohen in the next article
2917Keywords: Semimartingales with jumps,
Lévy processes,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
XXIX: 17, 181-193, LNM 1613 (1995)
COHEN, Serge
Some Markov properties of stochastic differential equations with jumps (
Stochastic differential geometry,
Markov processes)
The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see
1505 and
1655) was extended by Cohen to càdlàg semimartingales (
Stochastics Stochastics Rep. 56, 1996). Here this language is used to study the Markov property of solutions to SDE's with jumps. In particular,two definitions of a Lévy process in a Riemannian manifold are compared: One as the solution to a SDE driven by some Euclidean Lévy process, the other by subordinating some Riemannian Brownian motion. It is shown that in general the former is not of the second kind
Comment: The first definition is independently introduced by David Applebaum
2916Keywords: Semimartingales with jumps,
Lévy processes,
Subordination,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
XLV: 08, 201-244, LNM 2078 (2013)
CUCHIERO, Christa;
TEICHMANN, Josef
Path Properties and Regularity of Affine Processes on General State Spaces (
Theory of processes)
Keywords: affine processes,
path properties,
regularity,
Markov semimartingalesNature: Original
