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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 312

Keywords: Square integrable martingales, Angle bracket, Stochastic integrals

Nature: Exposition, Original additions

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II: 07, 123-139, LNM 51 (1968)

**SAM LAZARO, José de**

Sur les moments spectraux d'ordre supérieur (Second order processes)

The essential result of the paper (Shiryaev,*Th. Prob. Appl.*, **5**, 1960; Sinai, *Th. Prob. Appl.*, **8**, 1963) is the definition of multiple stochastic integrals with respect to a second order process whose covariance satisfies suitable spectral properties

Keywords: Spectral representation, Multiple stochastic integrals

Nature: Exposition

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III: 12, 160-162, LNM 88 (1969)

**MEYER, Paul-André**

Rectification à des exposés antérieurs (Markov processes, Martingale theory)

Corrections are given to the talk 202 by Cartier, Meyer and Weil and to the talk 106 by Meyer

Comment: This note introduces ``Walsh's fork'', the well-known strong Markov process whose dual is not strong Markov

Keywords: Time reversal, Stochastic integrals

Nature: Correction

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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 1017

Keywords: Local martingales, Stochastic integrals, Change of variable formula

Nature: Original

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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 integrals

Nature: Original

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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 representation

Nature: Original

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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 theorem

Nature: Exposition, Original additions

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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 integrals

Nature: Original

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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 integrals

Nature: Original

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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 systems

Nature: Original

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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 1530

Keywords: Stochastic integrals, Optional stochastic integrals, Random measures, Semimartingales

Nature: Original

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XI: 29, 418-434, LNM 581 (1977)

**LÉPINGLE, Dominique**

Sur la représentation des sauts des martingales (Martingale theory)

The problem discussed in this paper consists in decomposing into two parts a local martingale, so that one part has its jumps contained in a given thin optional set $D$ and the other one is continuous on $D$. The main theorem of 1121 is proved independently as an important technical tool

Comment: See also 1335

Keywords: Local martingales, Jumps, Optional stochastic integrals

Nature: Original

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XI: 30, 435-445, LNM 581 (1977)

**MAISONNEUVE, Bernard**

Une mise au point sur les martingales locales continues définies sur un intervalle stochastique (Martingale theory)

The following definition is given of a continuous local martingale $M$ on an open interval $[0,T[$, for an arbitrary stopping time $T$: two sequences are assumed to exist, one of stopping times $T_n\uparrow T$, one $(M_n)$ of continuous martingales, such that $M=M_n$ on $[0,T_n[$. Stochastic integration is studied, and the change of variable formula is extended. It is proved that the set where the limit $M_{T-}$ exists and is finite is a.s. the same as that where $\langle M,M\rangle_T<\infty$, a result whose proof under the usual definition (i.e., assuming $T$ is previsible) was not clear

Keywords: Martingales on a random set, Stochastic integrals

Nature: Original

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XI: 31, 446-481, LNM 581 (1977)

**MEYER, Paul-André**

Notes sur les intégrales stochastiques (Martingale theory)

This paper contains six additions to 1017. Chapter~I concerns Hilbert space valued martingales, following Métivier, defining in particular their operator valued brackets and the corresponding stochastic integrals. Chapter~II gives a new proof (due to Yan, and now classical) of the basic result on the structure of local martingales. Chapter~III is a theorem of Herz (and Lépingle in continuous time) on the representation of $BMO$ which corresponds to the ``maximal'' definition of $H^1$. Chapter~IV states that, if $(B_t)$ is a $BMO$ martingale and $(X_t)$ is a martingale bounded in $L^p$, then $\sup_t X^{\ast}_t |B_{\infty}-B_t|$ is also in $L^p$ with a norm controlled by that of $X$ ($1< p<\infty$; there is at least a wrong statement about $p=1$ at the bottom of p. 470). This result can be interpreted as $L^p$ boundedness of the commutator of two operators: multiplication by an element of $BMO$, and stochastic integration by a bounded previsible process. Chapter~V (again on $BMO$) has a wrong proof, and seems to be still an open problem. Chapter~VI consists of small additions and corrections, and in particular acknowledges the priority of P.W.~Millar for useful results on local times

Comment: Three errors are corrected in 1248 and 1249

Keywords: Stochastic integrals, Hilbert space valued martingales, Operator stochastic integrals, $BMO$

Nature: Original

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XI: 36, 518-528, LNM 581 (1977)

**YOR, Marc**

Sur quelques approximations d'intégrales stochastiques (Martingale theory)

The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form

Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe*Publ. RIMS, Kyoto Univ.* **15**, 1979; Bismut, Mécanique Aléatoire, Springer LNM~866, 1981; Meyer 1505

Keywords: Stochastic integrals, Riemann sums, Stratonovich integrals

Nature: Original

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XII: 20, 180-264, LNM 649 (1978)

**BISMUT, Jean-Michel**

Contrôle des systèmes linéaires quadratiques~: applications de l'intégrale stochastique (Control theory)

(To be completed) This is a set of lectures in control theory, which makes use of refined techniques in stochastic integration. It should be reviewed in detail

Comment: To be completed

Keywords: Optional stochastic integrals

Nature: Original

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XII: 48, 739-739, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Retour sur la représentation de $BMO$'' (Martingale theory)

Two errors in 1131 are corrected

Keywords: Stochastic integrals, $BMO$

Nature: Correction

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XII: 49, 739-739, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Caractérisation de $BMO$ par un opérateur maximal'' (Martingale theory)

Corrects an error in 1131

Keywords: Stochastic integrals, $BMO$

Nature: Correction

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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 1132

Keywords: Stochastic integrals

Nature: Correction

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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 spaces

Nature: Original

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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 integrals

Nature: Original

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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 integrals

Nature: Exposition

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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 inequality

Nature: Original

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XIII: 36, 427-440, LNM 721 (1979)

**YOR, Marc**

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (Brownian motion)

The problem is to study the filtration generated by real valued stochastic integrals $Y=\int_0^t(AX_s, dX_s)$, where $X$ is a $n$-dimensional Brownian motion, $A$ is a $n\times n$-matrix, and $(\,,\,)$ is the scalar product. If $A$ is the identity matrix we thus get (squares of) Bessel processes. If $A$ is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of $A$. In particular, this dimension is $1$ if and only if the matrix is equivalent to $cI_r$, a diagonal with $r$ ones and $n-r$ zeros. This is also (even if the symmetry assumption is omitted) the only case where $Y$ has the previsible representation property

Comment: Additional results on the same subject appear in 1545 and in Malric*Ann. Inst. H. Poincaré * **26** (1990)

Keywords: Stochastic integrals

Nature: Original

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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 1425

Keywords: Semimartingales, Stochastic integrals

Nature: Exposition

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XIV: 03, 18-25, LNM 784 (1980)

**CAIROLI, Renzo**

Sur l'extension de la définition d'intégrale stochastique (Several parameter processes)

A result of Wong-Zakai (*Ann. Prob.* **5**, 1977) extending the definition of the two kinds of stochastic integrals relative to the Brownian sheet is generalized to cover the case of stochastic integration relative to martingales, or strong martingales

Comment: A note at the end of the paper suggests some improvements

Keywords: Stochastic integrals, Brownian sheet

Nature: Original

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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 1533

Keywords: Stochastic integrals

Nature: Exposition, Original additions

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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 1415

Comment: Another approach to those integrals is due to L. Schwartz, in his article 1530 on formal semimartingales

Keywords: Stochastic integrals

Nature: Original

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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, Compensators

Nature: Original

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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 integrals

Nature: Original

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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 inequality

Nature: Exposition

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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 functions

Nature: Original

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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 1335

Keywords: Semimartingales, Optional stochastic integrals

Nature: Original

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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 1529

Keywords: Stochastic integrals

Nature: Original

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XIV: 47, 489-495, LNM 784 (1980)

**CAIROLI, Renzo**

Intégrale stochastique curviligne le long d'une courbe rectifiable (Several parameter processes)

The problem is to define stochastic integrals $\int_{\partial A} \phi\,\partial_1W$ where $W$ is the Brownian sheet, $\phi$ is a suitable process, and $A$ a suitable domain of the plane with rectifiable boundary

Keywords: Stochastic integrals, Brownian sheet

Nature: Original

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XV: 01, 1-5, LNM 850 (1981)

**FERNIQUE, Xavier**

Sur les lois de certaines intégrales associées à des mouvements browniens (Brownian motion)

Let $(Z_n)$ be a sequence of independent standard Brownian motions. Define by induction a sequence of processes $U_k$ by $U_0=Z_0$, $U_k(t)=\int_0^tU_{k-1}(s)dZ_k(s)$. Let $g_k(x)$ be the density of the random variable $U_k(1)$. Then the decrease at infinity of $g_k(x)$ is of the order $\exp(-C|x|^{\alpha})$ with $\alpha=2/(k+1)$ (slightly incorrect statement, see the paper for details)

Keywords: Iterated stochastic integrals

Nature: Original

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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 integrals

Nature: Original

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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 variation

Nature: Original

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XV: 28, 388-398, LNM 850 (1981)

**SPILIOTIS, Jean**

Sur les travaux de Krylov en théorie de l'intégrale stochastique (Martingale theory)

The well-known work of Malliavin deals with the existence of smooth densities for solutions of stochastic differential equations with smooth coefficients satisfying a hypoellipticity condition. N.V.~Krylov's earlier work (among many papers see*Izvestija Akad Nauk,* **38**, 1974, and Krylov's book *Controlled Diffusion processes,* Springer 1980) dealt with the existence of densities for several dimensional stochastic integrals with measurable bounded integrands, satisfying an ellipticity condition. It is a puzzling fact that nobody ever succeeded in unifying these results. Krylov's method depends on results of the Russian school on Monge-Ampère equations (see Pogorelov *The Minkowski Multidimensional Problem,* 1978). This exposition attempts, rather modestly, to explain in the seminar's language what it is all about, and in particular to show the place where a crucial lemma on convex functions is used

Keywords: Stochastic integrals, Existence of densities

Nature: Exposition

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 1428

Keywords: Stochastic integrals

Nature: 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 integrals

Nature: 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 inequality

Nature: Original

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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 1110

Keywords: Stopping times, Optional processes, Weak convergence, Stochastic integrals

Nature: Original

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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 processes

Nature: Correction

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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 semigroup

Nature: Original

Retrieve article from Numdam

XLIII: 07, 191-214, LNM 2006 (2011)

**RIEDLE, Markus**

Cylindrical Wiener Processes

Keywords: Cylindrical Wiener process, Cylindrical process, Cylindrical measure, Stochastic integrals, Stochastic differential equations, Radonifying operator, Reproducing kernel Hilbert space

Nature: Original

XLIII: 12, 309-325, LNM 2006 (2011)

**TUDOR, Ciprian A.**

Asymptotic Cramér's theorem and analysis on Wiener space (Limit theorems, Stochastic analysis)

Keywords: Multiple stochastic integrals, Limit theorems, Malliavin calculus, Stein's method

Nature: Original

XLIII: 18, 413-436, LNM 2006 (2011)

**CZICHOWSKY, Christoph**; **SCHWEIZER, Martin**

Closedness in the semimartingale topology for spaces of stochastic integrals with constrained integrands

Keywords: Stochastic integrals, Constrained strategies, Semimartingale topology, Closedness, Predictably convex, Projection on predictable range, Predictable correspondence, Optimisation under constraints, Mathematical finance

Nature: Original

XLV: 13, 323-351, LNM 2078 (2013)

**BOURGUIN, Solesne**; **TUDOR, Ciprian A.**

Malliavin Calculus and Self Normalized Sums (Theory of processes)

Keywords: Malliavin calculus, Stein's method, self-normalized sums, limit theorems, multiple stochastic integrals, chaos expansions

Nature: Original

Intégrales stochastiques I--IV (4 talks) (Martingale theory, Stochastic calculus)

This series presents an expanded exposition of the celebrated paper of Kunita-Watanabe (

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 312

Keywords: Square integrable martingales, Angle bracket, Stochastic integrals

Nature: Exposition, Original additions

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II: 07, 123-139, LNM 51 (1968)

Sur les moments spectraux d'ordre supérieur (Second order processes)

The essential result of the paper (Shiryaev,

Keywords: Spectral representation, Multiple stochastic integrals

Nature: Exposition

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III: 12, 160-162, LNM 88 (1969)

Rectification à des exposés antérieurs (Markov processes, Martingale theory)

Corrections are given to the talk 202 by Cartier, Meyer and Weil and to the talk 106 by Meyer

Comment: This note introduces ``Walsh's fork'', the well-known strong Markov process whose dual is not strong Markov

Keywords: Time reversal, Stochastic integrals

Nature: Correction

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IV: 09, 77-107, LNM 124 (1970)

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 1017

Keywords: Local martingales, Stochastic integrals, Change of variable formula

Nature: Original

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V: 14, 141-146, LNM 191 (1971)

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,

Keywords: Stochastic integrals

Nature: Original

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X: 03, 24-39, LNM 511 (1976)

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 representation

Nature: Original

Retrieve article from Numdam

X: 17, 245-400, LNM 511 (1976)

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 theorem

Nature: Exposition, Original additions

Retrieve article from Numdam

X: 19, 414-421, LNM 511 (1976)

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 integrals

Nature: Original

Retrieve article from Numdam

X: 20, 422-431, LNM 511 (1976)

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 integrals

Nature: Original

Retrieve article from Numdam

X: 22, 481-500, LNM 511 (1976)

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 systems

Nature: Original

Retrieve article from Numdam

XI: 26, 390-410, LNM 581 (1977)

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,

Keywords: Stochastic integrals, Optional stochastic integrals, Random measures, Semimartingales

Nature: Original

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XI: 29, 418-434, LNM 581 (1977)

Sur la représentation des sauts des martingales (Martingale theory)

The problem discussed in this paper consists in decomposing into two parts a local martingale, so that one part has its jumps contained in a given thin optional set $D$ and the other one is continuous on $D$. The main theorem of 1121 is proved independently as an important technical tool

Comment: See also 1335

Keywords: Local martingales, Jumps, Optional stochastic integrals

Nature: Original

Retrieve article from Numdam

XI: 30, 435-445, LNM 581 (1977)

Une mise au point sur les martingales locales continues définies sur un intervalle stochastique (Martingale theory)

The following definition is given of a continuous local martingale $M$ on an open interval $[0,T[$, for an arbitrary stopping time $T$: two sequences are assumed to exist, one of stopping times $T_n\uparrow T$, one $(M_n)$ of continuous martingales, such that $M=M_n$ on $[0,T_n[$. Stochastic integration is studied, and the change of variable formula is extended. It is proved that the set where the limit $M_{T-}$ exists and is finite is a.s. the same as that where $\langle M,M\rangle_T<\infty$, a result whose proof under the usual definition (i.e., assuming $T$ is previsible) was not clear

Keywords: Martingales on a random set, Stochastic integrals

Nature: Original

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XI: 31, 446-481, LNM 581 (1977)

Notes sur les intégrales stochastiques (Martingale theory)

This paper contains six additions to 1017. Chapter~I concerns Hilbert space valued martingales, following Métivier, defining in particular their operator valued brackets and the corresponding stochastic integrals. Chapter~II gives a new proof (due to Yan, and now classical) of the basic result on the structure of local martingales. Chapter~III is a theorem of Herz (and Lépingle in continuous time) on the representation of $BMO$ which corresponds to the ``maximal'' definition of $H^1$. Chapter~IV states that, if $(B_t)$ is a $BMO$ martingale and $(X_t)$ is a martingale bounded in $L^p$, then $\sup_t X^{\ast}_t |B_{\infty}-B_t|$ is also in $L^p$ with a norm controlled by that of $X$ ($1< p<\infty$; there is at least a wrong statement about $p=1$ at the bottom of p. 470). This result can be interpreted as $L^p$ boundedness of the commutator of two operators: multiplication by an element of $BMO$, and stochastic integration by a bounded previsible process. Chapter~V (again on $BMO$) has a wrong proof, and seems to be still an open problem. Chapter~VI consists of small additions and corrections, and in particular acknowledges the priority of P.W.~Millar for useful results on local times

Comment: Three errors are corrected in 1248 and 1249

Keywords: Stochastic integrals, Hilbert space valued martingales, Operator stochastic integrals, $BMO$

Nature: Original

Retrieve article from Numdam

XI: 36, 518-528, LNM 581 (1977)

Sur quelques approximations d'intégrales stochastiques (Martingale theory)

The investigation concerns the limit of several families of Riemann sums, converging to the Ito stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form

Comment: Stratonovich stochastic integrals of differential forms have been extensively studied in the context of stochastic differential geometry: see among others Ikeda-Manabe

Keywords: Stochastic integrals, Riemann sums, Stratonovich integrals

Nature: Original

Retrieve article from Numdam

XII: 20, 180-264, LNM 649 (1978)

Contrôle des systèmes linéaires quadratiques~: applications de l'intégrale stochastique (Control theory)

(To be completed) This is a set of lectures in control theory, which makes use of refined techniques in stochastic integration. It should be reviewed in detail

Comment: To be completed

Keywords: Optional stochastic integrals

Nature: Original

Retrieve article from Numdam

XII: 48, 739-739, LNM 649 (1978)

Correction à ``Retour sur la représentation de $BMO$'' (Martingale theory)

Two errors in 1131 are corrected

Keywords: Stochastic integrals, $BMO$

Nature: Correction

Retrieve article from Numdam

XII: 49, 739-739, LNM 649 (1978)

Correction à ``Caractérisation de $BMO$ par un opérateur maximal'' (Martingale theory)

Corrects an error in 1131

Keywords: Stochastic integrals, $BMO$

Nature: Correction

Retrieve article from Numdam

XII: 51, 740-740, LNM 649 (1978)

Correction à ``Sur un théorème de C. Stricker'' (Stochastic calculus)

Fills a gap in a proof in 1132

Keywords: Stochastic integrals

Nature: Correction

Retrieve article from Numdam

XII: 56, 757-762, LNM 649 (1978)

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 spaces

Nature: Original

Retrieve article from Numdam

XIII: 22, 250-252, LNM 721 (1979)

Caractérisation d'une classe de semimartingales (Martingale theory, Stochastic calculus)

The class of semimartingales $X$ such that the stochastic integral $J\,

Comment: This class has found recently a natural use in mathematical finance (Delbaen-Schachermayer 1997)

Keywords: Local martingales, Stochastic integrals

Nature: Original

Retrieve article from Numdam

XIII: 23, 253-259, LNM 721 (1979)

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 (

Keywords: Stochastic integrals

Nature: Exposition

Retrieve article from Numdam

XIII: 35, 407-426, LNM 721 (1979)

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 inequality

Nature: Original

Retrieve article from Numdam

XIII: 36, 427-440, LNM 721 (1979)

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (Brownian motion)

The problem is to study the filtration generated by real valued stochastic integrals $Y=\int_0^t(AX_s, dX_s)$, where $X$ is a $n$-dimensional Brownian motion, $A$ is a $n\times n$-matrix, and $(\,,\,)$ is the scalar product. If $A$ is the identity matrix we thus get (squares of) Bessel processes. If $A$ is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of $A$. In particular, this dimension is $1$ if and only if the matrix is equivalent to $cI_r$, a diagonal with $r$ ones and $n-r$ zeros. This is also (even if the symmetry assumption is omitted) the only case where $Y$ has the previsible representation property

Comment: Additional results on the same subject appear in 1545 and in Malric

Keywords: Stochastic integrals

Nature: Original

Retrieve article from Numdam

XIII: 54, 620-623, LNM 721 (1979)

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,

Comment: An important lemma which simplifies the proof and has other applications is given by Yan in 1425

Keywords: Semimartingales, Stochastic integrals

Nature: Exposition

Retrieve article from Numdam

XIV: 03, 18-25, LNM 784 (1980)

Sur l'extension de la définition d'intégrale stochastique (Several parameter processes)

A result of Wong-Zakai (

Comment: A note at the end of the paper suggests some improvements

Keywords: Stochastic integrals, Brownian sheet

Nature: Original

Retrieve article from Numdam

XIV: 15, 128-139, LNM 784 (1980)

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,

Keywords: Stochastic integrals

Nature: Exposition, Original additions

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XIV: 17, 148-151, LNM 784 (1980)

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 1415

Comment: Another approach to those integrals is due to L. Schwartz, in his article 1530 on formal semimartingales

Keywords: Stochastic integrals

Nature: Original

Retrieve article from Numdam

XIV: 18, 152-160, LNM 784 (1980)

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, Compensators

Nature: Original

Retrieve article from Numdam

XIV: 19, 161-172, LNM 784 (1980)

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 integrals

Nature: Original

Retrieve article from Numdam

XIV: 24, 209-219, LNM 784 (1980)

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

Keywords: Stochastic integrals, Stochastic differential equations, Métivier-Pellaumail inequality

Nature: Exposition

Retrieve article from Numdam

XIV: 25, 220-222, LNM 784 (1980)

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 functions

Nature: Original

Retrieve article from Numdam

XIV: 26, 223-226, LNM 784 (1980)

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 1335

Keywords: Semimartingales, Optional stochastic integrals

Nature: Original

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XIV: 28, 249-253, LNM 784 (1980)

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 (

Comment: See also 1529

Keywords: Stochastic integrals

Nature: Original

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XIV: 47, 489-495, LNM 784 (1980)

Intégrale stochastique curviligne le long d'une courbe rectifiable (Several parameter processes)

The problem is to define stochastic integrals $\int_{\partial A} \phi\,\partial_1W$ where $W$ is the Brownian sheet, $\phi$ is a suitable process, and $A$ a suitable domain of the plane with rectifiable boundary

Keywords: Stochastic integrals, Brownian sheet

Nature: Original

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XV: 01, 1-5, LNM 850 (1981)

Sur les lois de certaines intégrales associées à des mouvements browniens (Brownian motion)

Let $(Z_n)$ be a sequence of independent standard Brownian motions. Define by induction a sequence of processes $U_k$ by $U_0=Z_0$, $U_k(t)=\int_0^tU_{k-1}(s)dZ_k(s)$. Let $g_k(x)$ be the density of the random variable $U_k(1)$. Then the decrease at infinity of $g_k(x)$ is of the order $\exp(-C|x|^{\alpha})$ with $\alpha=2/(k+1)$ (slightly incorrect statement, see the paper for details)

Keywords: Iterated stochastic integrals

Nature: Original

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XV: 05, 44-102, LNM 850 (1981)

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

Comment: A short introduction by the same author can be found in

Keywords: Semimartingales in manifolds, Martingales in manifolds, Transfer principle, Stochastic differential equations, Stochastic integrals, Stratonovich integrals

Nature: Original

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XV: 09, 143-150, LNM 850 (1981)

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

Keywords: Stochastic integrals, Change of variable formula, Quadratic variation

Nature: Original

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XV: 28, 388-398, LNM 850 (1981)

Sur les travaux de Krylov en théorie de l'intégrale stochastique (Martingale theory)

The well-known work of Malliavin deals with the existence of smooth densities for solutions of stochastic differential equations with smooth coefficients satisfying a hypoellipticity condition. N.V.~Krylov's earlier work (among many papers see

Keywords: Stochastic integrals, Existence of densities

Nature: Exposition

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XV: 29, 399-412, LNM 850 (1981)

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 (

Keywords: Stochastic integrals

Nature: Original

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XV: 30, 413-489, LNM 850 (1981)

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,

Keywords: Semimartingales, Formal semimartingales, Stochastic integrals

Nature: Original

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XV: 33, 499-522, LNM 850 (1981)

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 inequality

Nature: Original

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XVI: 27, 314-318, LNM 920 (1982)

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 1110

Keywords: Stopping times, Optional processes, Weak convergence, Stochastic integrals

Nature: Original

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XX: 02, 28-29, LNM 1204 (1986)

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 processes

Nature: Correction

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XXVI: 18, 189-209, LNM 1526 (1992)

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 semigroup

Nature: Original

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XLIII: 07, 191-214, LNM 2006 (2011)

Cylindrical Wiener Processes

Keywords: Cylindrical Wiener process, Cylindrical process, Cylindrical measure, Stochastic integrals, Stochastic differential equations, Radonifying operator, Reproducing kernel Hilbert space

Nature: Original

XLIII: 12, 309-325, LNM 2006 (2011)

Asymptotic Cramér's theorem and analysis on Wiener space (Limit theorems, Stochastic analysis)

Keywords: Multiple stochastic integrals, Limit theorems, Malliavin calculus, Stein's method

Nature: Original

XLIII: 18, 413-436, LNM 2006 (2011)

Closedness in the semimartingale topology for spaces of stochastic integrals with constrained integrands

Keywords: Stochastic integrals, Constrained strategies, Semimartingale topology, Closedness, Predictably convex, Projection on predictable range, Predictable correspondence, Optimisation under constraints, Mathematical finance

Nature: Original

XLV: 13, 323-351, LNM 2078 (2013)

Malliavin Calculus and Self Normalized Sums (Theory of processes)

Keywords: Malliavin calculus, Stein's method, self-normalized sums, limit theorems, multiple stochastic integrals, chaos expansions

Nature: Original