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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: 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: 25, 383-389, LNM 581 (1977)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

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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XII: 05, 47-50, LNM 649 (1978)

**KAZAMAKI, Norihiko**

A remark on a problem of Girsanov (Martingale theory)

It is shown that, if $M$ is a continuous local martingale which belongs to $BMO$, its stochastic exponential is a uniformly integrable martingale

Comment: This has become a well-known result. It is false for complex valued martingales, even bounded ones: see 1832

Keywords: Stochastic exponentials, $BMO$

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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XVIII: 32, 500-500, LNM 1059 (1984)

**ÉMERY, Michel**

Sur l'exponentielle d'une martingale de $BMO$ (Martingale theory)

This very short note remarks that for complex-valued processes, it is no longer true that the stochastic exponential of a bounded martingale is a martingale---it is only a local martingale

Keywords: Stochastic exponentials, $BMO$

Nature: Original

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XLIV: 02, 41-59, LNM 2046 (2012)

**MIJATOVIĆ, Aleksandar**; **NOVAK, Nika**; **URUSOV, Mikhail**

Martingale property of generalized stochastic exponentials (Theory of martingales)

Keywords: Generalized stochastic exponentials, Local martingales vs. true martingales, One-dimensional diffusions

Nature: Original

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)

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: 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: 25, 383-389, LNM 581 (1977)

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

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

XII: 05, 47-50, LNM 649 (1978)

A remark on a problem of Girsanov (Martingale theory)

It is shown that, if $M$ is a continuous local martingale which belongs to $BMO$, its stochastic exponential is a uniformly integrable martingale

Comment: This has become a well-known result. It is false for complex valued martingales, even bounded ones: see 1832

Keywords: Stochastic exponentials, $BMO$

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

XVIII: 32, 500-500, LNM 1059 (1984)

Sur l'exponentielle d'une martingale de $BMO$ (Martingale theory)

This very short note remarks that for complex-valued processes, it is no longer true that the stochastic exponential of a bounded martingale is a martingale---it is only a local martingale

Keywords: Stochastic exponentials, $BMO$

Nature: Original

Retrieve article from Numdam

XLIV: 02, 41-59, LNM 2046 (2012)

Martingale property of generalized stochastic exponentials (Theory of martingales)

Keywords: Generalized stochastic exponentials, Local martingales vs. true martingales, One-dimensional diffusions

Nature: Original