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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: 15, 134-137, LNM 649 (1978)

**LÉPINGLE, Dominique**

Une inégalité de martingales (Martingale theory)

The following inequality for a discrete time adapted process $(a_n)$ and its conditional expectations $b_n=E[a_n\,|\,{\cal F}_{n-1}]$ is proved: $$\|(\sum_n b_n^2)^{1/2}\|_1\le 2\|(\sum_n a_n^2)^{1/2}\|_1\ .$$ A similar inequality in $L^p$, $1\!<\!p\!<\!\infty$, does not require adaptedness, and is due to Stein

Keywords: Inequalities, Quadratic variation

Nature: Original

Retrieve article from Numdam

XII: 16, 138-147, LNM 649 (1978)

**LÉPINGLE, Dominique**

Sur certains commutateurs de la théorie des martingales (Martingale theory)

Let $\beta$ the operator on (closed) martingales $X$ consisting in multiplication of $X_{\infty}$ by a given r.v. $B$. One investigates the commutator $J\beta-\beta J$ of $\beta$ with some operator $J$ on martingales (a typical example is stochastic integration $JX=H.X$ where $H$ is a given bounded previsible process), expecting this commutator to be bounded in $L^p$ if $B$ belongs to $BMO$. This is indeed true under natural conditions on $J$

Keywords: $BMO$

Nature: Original

Retrieve article from Numdam

XII: 17, 148-161, LNM 649 (1978)

**LÉPINGLE, Dominique**

Sur le comportement asymptotique des martingales locales (Martingale theory)

This paper is devoted to the extension of well-known statements (the strong law of large numbers, the Borel-Cantelli lemma, and the easier half of the law of the iterated logarithm) to right-continuous local martingales. An interesting technical point is the definition of a family of exponential supermartingales

Keywords: Law of large numbers, Borel-Cantelli lemma, Exponential martingales, Law of the iterated logarithm

Nature: Original

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XII: 50, 739-739, LNM 649 (1978)

**LÉPINGLE, Dominique**

Correction au Séminaire X (Martingale theory)

Corrects a detail in 1018

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Correction

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XIII: 26, 294-306, LNM 721 (1979)

**BONAMI, Aline**; **LÉPINGLE, Dominique**

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $**P**$ two functionals of a $**P**$-martingale---remains true, possibly with new constants, when $**P**$ is replaced by an equivalent law $Z.**P**$. To this order, the ``weight'' $Z$ must satisfy special conditions, among which a probabilistic version of Muckenhoupt's (1972) $(A_p)$ condition and a condition of multiplicative boundedness on the jumps of the martingale $E[Z\,|\,{\cal F}_t]$. This volume contains three papers on weighted norms inequalities, 1326, 1327, 1328, with considerable overlap. Here the main topic is the weighted-norm extension of the Burkholder-Gundy inequalities

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

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XIV: 04, 26-48, LNM 784 (1980)

**LENGLART, Érik**; **LÉPINGLE, Dominique**; **PRATELLI, Maurizio**

Présentation unifiée de certaines inégalités de la théorie des martingales (Martingale theory)

This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes

Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,*Probabilités et Potentiel E *; see also 1621. A striking example of the power of these methods is Barlow-Yor, {\sl Jour. Funct. Anal.} **49**,1982

Keywords: Moderate convex functions, Inequalities, Martingale inequalities, Burkholder inequalities, Good lambda inequalities, Domination inequalities

Nature: Original

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XIV: 22, 200-204, LNM 784 (1980)

**AUERHAN, J.**; **LÉPINGLE, Dominique**; **YOR, Marc**

Construction d'une martingale réelle continue de filtration naturelle donnée (General theory of processes)

It is proved in 1123 that, under a mere separability assumption, any filtration is the natural filtration of some bounded left-continuous increasing process $(A_t)$. If the filtration contains a Brownian motion $(B_t)$, then it is also the natural filtration of the continuous martingale $\int_0^t A_sdB_s$. Therefore, the natural filtration of (finitely or countably) many independent Brownian motions is generated by a single continuous martingale. Explicit constructions are discussed

Nature: Original

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XV: 41, 604-617, LNM 850 (1981)

**LÉPINGLE, Dominique**; **MEYER, Paul-André**; **YOR, Marc**

Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)

This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case

Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps

Nature: Original

Retrieve article from Numdam

XV: 45, 643-668, LNM 850 (1981)

**AUERHAN, J.**; **LÉPINGLE, Dominique**

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (General theory of processes, Brownian motion, Martingale theory)

This is a sequel to 1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in 1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given

Comment: Further results are given by Malric*Ann. Inst. H. Poincaré * **26** (1990)

Nature: Original

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XV: 46, 669-670, LNM 850 (1981)

**LÉPINGLE, Dominique**

Une remarque sur les lois de certains temps d'atteinte (Brownian motion)

Let $T$ be the exit time of the interval $[-d,c]$ for a Brownian motion starting at $0$. A classical formula giving the Laplace transform of the law of $T$ can be extended by analytical continuation to the positive axis. It is shown here that this extension has a purely probabilistic proof. The same method gives two other formulas

Keywords: Exit time from an interval

Nature: New proof of known results

Retrieve article from Numdam

XX: 23, 352-374, LNM 1204 (1986)

**HAKIM-DOWEK, M.**; **LÉPINGLE, Dominique**

L'exponentielle stochastique des groupes de Lie (Stochastic differential geometry)

Given a Lie group $G$ and its Lie algebra $\cal G$, this article defines and studies the stochastic exponential of a (continuous) semimartingale $M$ in $\cal G$ as the solution in $G$ to the Stratonovich s.d.e. $dX = X dM$. The inverse operation (stochastic logarithm) is also considered; various formulas are established (e.g. the exponential of $M+N$). When $M$ is a local martingale, $X$ is a martingale for the connection such that $\nabla_A B=0$ for all left-invariant vector fields $A$ and $B$

Comment: See also Karandikar*Ann. Prob.* **10** (1982) and 1722. For a sequel, see Arnaudon 2612

Keywords: Semimartingales in manifolds, Martingales in manifolds, Lie group

Nature: Original

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XXI: 30, 520-533, LNM 1247 (1987)

**LÉPINGLE, Dominique**; **MAROIS, Christine**

Équations différentielles stochastiques multivoques unidimensionnelles

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XXVI: 16, 167-169, LNM 1526 (1992)

**LÉPINGLE, Dominique**

Orthogonalité et équiintégrabilité des martingales discrètes

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XL: 12, 241-246, LNM 1899 (2007)

**CÉPA, Emmanuel**; **LÉPINGLE, Dominique**

No multiple collisions for mutually repelling Brownian particles

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: 15, 134-137, LNM 649 (1978)

Une inégalité de martingales (Martingale theory)

The following inequality for a discrete time adapted process $(a_n)$ and its conditional expectations $b_n=E[a_n\,|\,{\cal F}_{n-1}]$ is proved: $$\|(\sum_n b_n^2)^{1/2}\|_1\le 2\|(\sum_n a_n^2)^{1/2}\|_1\ .$$ A similar inequality in $L^p$, $1\!<\!p\!<\!\infty$, does not require adaptedness, and is due to Stein

Keywords: Inequalities, Quadratic variation

Nature: Original

Retrieve article from Numdam

XII: 16, 138-147, LNM 649 (1978)

Sur certains commutateurs de la théorie des martingales (Martingale theory)

Let $\beta$ the operator on (closed) martingales $X$ consisting in multiplication of $X_{\infty}$ by a given r.v. $B$. One investigates the commutator $J\beta-\beta J$ of $\beta$ with some operator $J$ on martingales (a typical example is stochastic integration $JX=H.X$ where $H$ is a given bounded previsible process), expecting this commutator to be bounded in $L^p$ if $B$ belongs to $BMO$. This is indeed true under natural conditions on $J$

Keywords: $BMO$

Nature: Original

Retrieve article from Numdam

XII: 17, 148-161, LNM 649 (1978)

Sur le comportement asymptotique des martingales locales (Martingale theory)

This paper is devoted to the extension of well-known statements (the strong law of large numbers, the Borel-Cantelli lemma, and the easier half of the law of the iterated logarithm) to right-continuous local martingales. An interesting technical point is the definition of a family of exponential supermartingales

Keywords: Law of large numbers, Borel-Cantelli lemma, Exponential martingales, Law of the iterated logarithm

Nature: Original

Retrieve article from Numdam

XII: 50, 739-739, LNM 649 (1978)

Correction au Séminaire X (Martingale theory)

Corrects a detail in 1018

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Correction

Retrieve article from Numdam

XIII: 26, 294-306, LNM 721 (1979)

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

Retrieve article from Numdam

XIV: 04, 26-48, LNM 784 (1980)

Présentation unifiée de certaines inégalités de la théorie des martingales (Martingale theory)

This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes

Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,

Keywords: Moderate convex functions, Inequalities, Martingale inequalities, Burkholder inequalities, Good lambda inequalities, Domination inequalities

Nature: Original

Retrieve article from Numdam

XIV: 22, 200-204, LNM 784 (1980)

Construction d'une martingale réelle continue de filtration naturelle donnée (General theory of processes)

It is proved in 1123 that, under a mere separability assumption, any filtration is the natural filtration of some bounded left-continuous increasing process $(A_t)$. If the filtration contains a Brownian motion $(B_t)$, then it is also the natural filtration of the continuous martingale $\int_0^t A_sdB_s$. Therefore, the natural filtration of (finitely or countably) many independent Brownian motions is generated by a single continuous martingale. Explicit constructions are discussed

Nature: Original

Retrieve article from Numdam

XV: 41, 604-617, LNM 850 (1981)

Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)

This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case

Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps

Nature: Original

Retrieve article from Numdam

XV: 45, 643-668, LNM 850 (1981)

Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (II) (General theory of processes, Brownian motion, Martingale theory)

This is a sequel to 1336. The problem is to describe the filtration of the continuous martingale $\int_0^t (AX_s,dX_s)$ where $X$ is a $n$-dimensional Brownian motion. It is shown that if the matrix $A$ is normal (rather than symmetric as in 1336) then this filtration is that of a (several dimensional) Brownian motion. If $A$ is not normal, only a lower bound on the multiplicity of this filtration can be given, and the problem is far from solved. The complex case is also considered. Several examples are given

Comment: Further results are given by Malric

Nature: Original

Retrieve article from Numdam

XV: 46, 669-670, LNM 850 (1981)

Une remarque sur les lois de certains temps d'atteinte (Brownian motion)

Let $T$ be the exit time of the interval $[-d,c]$ for a Brownian motion starting at $0$. A classical formula giving the Laplace transform of the law of $T$ can be extended by analytical continuation to the positive axis. It is shown here that this extension has a purely probabilistic proof. The same method gives two other formulas

Keywords: Exit time from an interval

Nature: New proof of known results

Retrieve article from Numdam

XX: 23, 352-374, LNM 1204 (1986)

L'exponentielle stochastique des groupes de Lie (Stochastic differential geometry)

Given a Lie group $G$ and its Lie algebra $\cal G$, this article defines and studies the stochastic exponential of a (continuous) semimartingale $M$ in $\cal G$ as the solution in $G$ to the Stratonovich s.d.e. $dX = X dM$. The inverse operation (stochastic logarithm) is also considered; various formulas are established (e.g. the exponential of $M+N$). When $M$ is a local martingale, $X$ is a martingale for the connection such that $\nabla_A B=0$ for all left-invariant vector fields $A$ and $B$

Comment: See also Karandikar

Keywords: Semimartingales in manifolds, Martingales in manifolds, Lie group

Nature: Original

Retrieve article from Numdam

XXI: 30, 520-533, LNM 1247 (1987)

Équations différentielles stochastiques multivoques unidimensionnelles

Retrieve article from Numdam

XXVI: 16, 167-169, LNM 1526 (1992)

Orthogonalité et équiintégrabilité des martingales discrètes

Retrieve article from Numdam

XL: 12, 241-246, LNM 1899 (2007)

No multiple collisions for mutually repelling Brownian particles