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VIII: 17, 310-315, LNM 381 (1974)

**MEYER, Paul-André**

Une représentation de surmartingales (Martingale theory)

Garsia asked whether every right continuous positive supermartingale $(X_t)$ bounded by $1$ is the optional projection of a (non-adapted) decreasing process $(D_t)$, also bounded by $1$. This problem is solved by an explicit formula, and a proof is sketched showing that, if boundedness is not assumed, the proper condition is $D_t\le X^{*}$

Comment: The ``exponential formula'' appearing in this paper was suggested by a more concrete problem in the theory of Markov processes, using a terminal time. Similar looking formulas occurs in multiplicative decompositions and in 801. For the much more difficult case of positive submartingales, see 1023 and above all Azéma,*Z. für W-theorie,* **45**, 1978 and its exposition 1321

Keywords: Supermartingales, Multiplicative decomposition

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: 21, 432-480, LNM 511 (1976)

**YOEURP, Chantha**

Décomposition des martingales locales et formules exponentielles (Martingale theory, Stochastic calculus)

It is shown that local martingales can be decomposed uniquely into three pieces, a continuous part and two purely discontinuous pieces, one with accessible jumps, and one with totally inaccessible jumps. Two beautiful lemmas say that a purely discontinuous local martingale whose jumps are summable is a finite variation process, and if it has accessible jumps, then it is the sum of its jumps without compensation. Conditions are given for the existence of the angle bracket of two local martingales which are not locally square integrable. Lemma 2.3 is the lemma often quoted as ``Yoeurp's Lemma'': given a local martingale $M$ and a previsible process of finite variation $A$, $[M,A]$ is a local martingale. The definition of a local martingale on an open interval $[0,T[$ is given when $T$ is previsible, and the behaviour of local martingales under changes of laws (Girsanov's theorem) is studied in a set up where the positive martingale defining the mutual density is replaced by a local martingale. The existence and uniqueness of solutions of the equation $Z_t=1+\int_0^t\tilde Z_s dX_s$, where $X$ is a given special semimartingale of decomposition $M+A$, and $\widetilde Z$ is the previsible projection of the unknown special semimartingale $Z$, is proved under an assumption that the jumps $ėlta A_t$ do not assume the value $1$. Then this ``exponential'' is used to study the multiplicative decomposition of a positive supermartingale in full generality

Comment: The problems in this paper have some relation with Kunita 1005 (in a Markovian set up), and are further studied by Yoeurp in LN**1118**, *Grossissements de filtrations,* 1985. The subject of multiplicative decompositions of positive submartingales is much more difficult since they may vanish. For a simple case see in this volume Yoeurp-Meyer 1023. The general case is due to Azéma (*Z. für W-theorie,* **45,** 1978, presented in 1321) See also 1622

Keywords: Stochastic exponentials, Multiplicative decomposition, Angle bracket, Girsanov's theorem, Föllmer measures

Nature: Original

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X: 23, 501-504, LNM 511 (1976)

**MEYER, Paul-André**; **YOEURP, Chantha**

Sur la décomposition multiplicative des sousmartingales positives (Martingale theory)

This paper expands part of Yoeurp's paper 1021, to cover the decomposition of positive submartingales instead supermartingales, assuming that the process never vanishes. A corollary is that every positive (not necessarily strictly so) submartingale $X_t$ is the optional projection of an increasing process $C_t$, non-adapted, such that $0\leq C_t\leq X_{\infty}$

Comment: See the comments on 1021 for the general case. The latter result is related to Meyer 817. For a related paper, see 1203. Further study in 1620

Keywords: Multiplicative decomposition

Nature: Original

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XII: 03, 22-34, LNM 649 (1978)

**JACOD, Jean**

Projection prévisible et décomposition multiplicative d'une semi-martingale positive (General theory of processes)

The problem discussed is the decomposition of a positive ($\ge0$) special semimartingale $X$ (the most interesting cases being super- and submartingales) into a product of a positive local martingale and a positive previsible process of finite variation. The problem is solved here in the greatest possible generality, on a maximal non-vanishing domain for $X$---this is a previsible stochastic interval $[0,S)$ which at $S$ may be open or closed

Comment: This papers improves on 1021 and 1023

Keywords: Semimartingales, Multiplicative decomposition

Nature: Original

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XII: 04, 35-46, LNM 649 (1978)

**MÉMIN, Jean**

Décompositions multiplicatives de semimartingales exponentielles et applications (General theory of processes)

It is shown that, given two semimartingales $U,V$ such that $U$ has no jump equal to $-1$, there is a unique semimartingale $X$ such that ${\cal E}(X)\,{\cal E}(U)={\cal E}(V)$. This result is applied to recover all known results on multiplicative decompositions

Comment: The results of this paper are used in Mémin-Shiryaev 1312

Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition

Nature: Original

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XIII: 12, 142-161, LNM 721 (1979)

**MÉMIN, Jean**; **SHIRYAEV, Albert N.**

Un critère prévisible pour l'uniforme intégrabilité des semimartingales exponentielles (Martingale theory)

A condition is given so that the stochastic exponential of a special semimartingale $X$ is a uniformly integrable process. It involves only the local characteristics of $X$, i.e., its previsible compensator, Lévy measure, and quadratic variation of the continuous martingale part. The proof rests on multiplicative decompositions, and known results in the case of martingales

Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition, Local characteristics

Nature: Original

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XIII: 21, 240-249, LNM 721 (1979)

**MEYER, Paul-André**

Représentations multiplicatives de sousmartingales, d'après Azéma (Martingale theory)

The problem of the multiplicative decomposition of a positive supermartingale $Y$ is relatively easy, but the similar problem for a positive submartingale (find a previsible decreasing process $(C_t)$ such that $(C_tY_t)$ is a martingale) is plagued by the zeros of $Y$. An important idea of Azéma (*Zeit. für W-Theorie,* **45**, 1978) is the introduction of a *multiplicative system * as a two-parameter process $C_{st}$ taking values in $[0,1]$, defined for $s\le t$, such that $C_{tt}=1$, $C_{st}C_{tu}=C_{su}$ for $s\le t\le u$, decreasing and previsible in $t$ for fixed $s$, such that $E[C_{st}Y_t\,|\,{\cal F}_s]=Y_s$ for $s<t$. Then for fixed $s$ the process $(C_{st}Y_t)$ turns out to be a right-continuous martingale on $[s,\infty[$, and what we have done amounts to pasting together all the multiplicative decompositions on zero-free intervals. Existence (and uniqueness of multiplicative systems are proved, though the uniqueness result is slightly different from Azéma's

Keywords: Multiplicative decomposition

Nature: Exposition, Original additions

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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

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Une représentation de surmartingales (Martingale theory)

Garsia asked whether every right continuous positive supermartingale $(X_t)$ bounded by $1$ is the optional projection of a (non-adapted) decreasing process $(D_t)$, also bounded by $1$. This problem is solved by an explicit formula, and a proof is sketched showing that, if boundedness is not assumed, the proper condition is $D_t\le X^{*}$

Comment: The ``exponential formula'' appearing in this paper was suggested by a more concrete problem in the theory of Markov processes, using a terminal time. Similar looking formulas occurs in multiplicative decompositions and in 801. For the much more difficult case of positive submartingales, see 1023 and above all Azéma,

Keywords: Supermartingales, Multiplicative decomposition

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: 21, 432-480, LNM 511 (1976)

Décomposition des martingales locales et formules exponentielles (Martingale theory, Stochastic calculus)

It is shown that local martingales can be decomposed uniquely into three pieces, a continuous part and two purely discontinuous pieces, one with accessible jumps, and one with totally inaccessible jumps. Two beautiful lemmas say that a purely discontinuous local martingale whose jumps are summable is a finite variation process, and if it has accessible jumps, then it is the sum of its jumps without compensation. Conditions are given for the existence of the angle bracket of two local martingales which are not locally square integrable. Lemma 2.3 is the lemma often quoted as ``Yoeurp's Lemma'': given a local martingale $M$ and a previsible process of finite variation $A$, $[M,A]$ is a local martingale. The definition of a local martingale on an open interval $[0,T[$ is given when $T$ is previsible, and the behaviour of local martingales under changes of laws (Girsanov's theorem) is studied in a set up where the positive martingale defining the mutual density is replaced by a local martingale. The existence and uniqueness of solutions of the equation $Z_t=1+\int_0^t\tilde Z_s dX_s$, where $X$ is a given special semimartingale of decomposition $M+A$, and $\widetilde Z$ is the previsible projection of the unknown special semimartingale $Z$, is proved under an assumption that the jumps $ėlta A_t$ do not assume the value $1$. Then this ``exponential'' is used to study the multiplicative decomposition of a positive supermartingale in full generality

Comment: The problems in this paper have some relation with Kunita 1005 (in a Markovian set up), and are further studied by Yoeurp in LN

Keywords: Stochastic exponentials, Multiplicative decomposition, Angle bracket, Girsanov's theorem, Föllmer measures

Nature: Original

Retrieve article from Numdam

X: 23, 501-504, LNM 511 (1976)

Sur la décomposition multiplicative des sousmartingales positives (Martingale theory)

This paper expands part of Yoeurp's paper 1021, to cover the decomposition of positive submartingales instead supermartingales, assuming that the process never vanishes. A corollary is that every positive (not necessarily strictly so) submartingale $X_t$ is the optional projection of an increasing process $C_t$, non-adapted, such that $0\leq C_t\leq X_{\infty}$

Comment: See the comments on 1021 for the general case. The latter result is related to Meyer 817. For a related paper, see 1203. Further study in 1620

Keywords: Multiplicative decomposition

Nature: Original

Retrieve article from Numdam

XII: 03, 22-34, LNM 649 (1978)

Projection prévisible et décomposition multiplicative d'une semi-martingale positive (General theory of processes)

The problem discussed is the decomposition of a positive ($\ge0$) special semimartingale $X$ (the most interesting cases being super- and submartingales) into a product of a positive local martingale and a positive previsible process of finite variation. The problem is solved here in the greatest possible generality, on a maximal non-vanishing domain for $X$---this is a previsible stochastic interval $[0,S)$ which at $S$ may be open or closed

Comment: This papers improves on 1021 and 1023

Keywords: Semimartingales, Multiplicative decomposition

Nature: Original

Retrieve article from Numdam

XII: 04, 35-46, LNM 649 (1978)

Décompositions multiplicatives de semimartingales exponentielles et applications (General theory of processes)

It is shown that, given two semimartingales $U,V$ such that $U$ has no jump equal to $-1$, there is a unique semimartingale $X$ such that ${\cal E}(X)\,{\cal E}(U)={\cal E}(V)$. This result is applied to recover all known results on multiplicative decompositions

Comment: The results of this paper are used in Mémin-Shiryaev 1312

Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition

Nature: Original

Retrieve article from Numdam

XIII: 12, 142-161, LNM 721 (1979)

Un critère prévisible pour l'uniforme intégrabilité des semimartingales exponentielles (Martingale theory)

A condition is given so that the stochastic exponential of a special semimartingale $X$ is a uniformly integrable process. It involves only the local characteristics of $X$, i.e., its previsible compensator, Lévy measure, and quadratic variation of the continuous martingale part. The proof rests on multiplicative decompositions, and known results in the case of martingales

Keywords: Stochastic exponentials, Semimartingales, Multiplicative decomposition, Local characteristics

Nature: Original

Retrieve article from Numdam

XIII: 21, 240-249, LNM 721 (1979)

Représentations multiplicatives de sousmartingales, d'après Azéma (Martingale theory)

The problem of the multiplicative decomposition of a positive supermartingale $Y$ is relatively easy, but the similar problem for a positive submartingale (find a previsible decreasing process $(C_t)$ such that $(C_tY_t)$ is a martingale) is plagued by the zeros of $Y$. An important idea of Azéma (

Keywords: Multiplicative decomposition

Nature: Exposition, Original additions

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