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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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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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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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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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

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

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

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

Retrieve article from Numdam