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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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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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XV: 07, 118-141, LNM 850 (1981)

**KUNITA, Hiroshi**

Some extensions of Ito's formula (Stochastic calculus)

The standard Ito formula expresses the composition of a smooth function $f$ with a continuous semimartingale as a stochastic integral, thus implying that the composition itself is a semimartingale. The extensions of Ito formula considered here deal with more complicated composition problems. The first one concerns a composition Let $(F(t, X_t)$ where $F(t,x)$ is a continuous semimartingale depending on a parameter $x\in**R**^d$ and satisfying convenient regularity assumptions, and $X_t$ is a semimartingale. Typically $F(t,x)$ will be the flow of diffeomorphisms arising from a s.d.e. with the initial point $x$ as variable. Other examples concern the parallel transport of tensors along the paths of a flow of diffeomorphisms, or the pull-back of a tensor field by the flow itself. Such formulas (developed also by Bismut) are very useful tools of stochastic differential geometry

Keywords: Stochastic differential equations, Flow of a s.d.e., Change of variable formula, Stochastic parallel transport

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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XVI: 20, 234-237, LNM 920 (1982)

**YOEURP, Chantha**

Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (Brownian motion, Stochastic calculus)

A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$

Comment: See 1023, 1321

Keywords: Multiplicative decomposition, Change of variable formula, Local times

Nature: Original

Retrieve article from Numdam

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

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

XV: 07, 118-141, LNM 850 (1981)

Some extensions of Ito's formula (Stochastic calculus)

The standard Ito formula expresses the composition of a smooth function $f$ with a continuous semimartingale as a stochastic integral, thus implying that the composition itself is a semimartingale. The extensions of Ito formula considered here deal with more complicated composition problems. The first one concerns a composition Let $(F(t, X_t)$ where $F(t,x)$ is a continuous semimartingale depending on a parameter $x\in

Keywords: Stochastic differential equations, Flow of a s.d.e., Change of variable formula, Stochastic parallel transport

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

XVI: 20, 234-237, LNM 920 (1982)

Une décomposition multiplicative de la valeur absolue d'un mouvement brownien (Brownian motion, Stochastic calculus)

A positive submartingale like $X_t=|B_t|$ vanishes too often to be represented as a product of a local martingale and an increasing process. Still, one may look for a kind of additive decomposition of $\log X$, from which the required multiplicative decomposition would follow by taking exponentials. Here the (Ito-Tanaka) additive decomposition of $\log(X\lor\epsilon)$ is studied, as well as its limiting behaviour as $\epsilon\rightarrow0$

Comment: See 1023, 1321

Keywords: Multiplicative decomposition, Change of variable formula, Local times

Nature: Original

Retrieve article from Numdam