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XI: 28, 415-417, LNM 581 (1977)

**LENGLART, Érik**

Une caractérisation des processus prévisibles (General theory of processes)

One of the results of this short paper is the following: a bounded optional process $X$ is previsible if and only if, for every martingale $M$ of integrable variation, the Stieltjes integral process $X\sc M$ is a martingale

Keywords: Previsible processes

Nature: Original

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XII: 07, 53-56, LNM 649 (1978)

**LENGLART, Érik**

Sur la localisation des intégrales stochastiques (Stochastic calculus)

A mapping $T$ from processes to processes is*local * if, whenever two processes $X,Y$ are equal on an event $A\subset\Omega$, the same is true for $TX,TY$. Classical results on locality in stochastic calculus are derived here in a simple way from the generalized Girsanov theorem (which concerns a pair of laws $P,Q$ with $Q$ absolutely continuous with respect to $P$, but not necessarily equivalent to it: see Lenglart, *Zeit. für W-theorie,* 39, 1977). A new result is derived: if $X$ and $Y$ are semimartingales and their difference is of finite variation on an event $A$, then their continuous martingale parts are equal on $A$

Keywords: Girsanov's theorem

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: 05, 49-52, LNM 784 (1980)

**LENGLART, Érik**

Appendice à l'exposé précédent~: inégalités de semimartingales (Martingale theory, Stochastic calculus)

This paper contains several applications of the methods of 1404 to the case of semimartingales instead of martingales

Keywords: Inequalities, Semimartingales

Nature: Original

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XIV: 14, 125-127, LNM 784 (1980)

**LENGLART, Érik**

Sur l'inégalité de Métivier-Pellaumail (Stochastic calculus)

A simplified (but still not so simple) proof of the Métivier-Pellaumail inequality

Keywords: Doob's inequality, Métivier-Pellaumail inequality

Nature: New proof of known results

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XIV: 49, 500-546, LNM 784 (1980)

**LENGLART, Érik**

Tribus de Meyer et théorie des processus (General theory of processes, Stochastic calculus)

The subject of this paper is the study of the $\sigma$-field on $**R**_+\times\Omega$ generated by a family of cadlag processes including the deterministic ones, and stable under stopping at non-random times. Of course the optional and previsible $\sigma$-fields are Meyer $\sigma$-fields in this very general sense. It is a matter of wonder to see how far one can go with such simple hypotheses, which were suggested by Dellacherie 705

Comment: This beautiful paper was generally ignored. If a suggestive name had been used instead of the terminology ``Meyer $\sigma$-field'', its fate might have been different. See 1524 for an interesting application. The work of Fourati (partly unpublished) follows along the same lines, but including time reversal: see 2119

Keywords: Projection theorems, Section theorems

Nature: Original

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XV: 24, 320-346, LNM 850 (1981)

**DELLACHERIE, Claude**; **LENGLART, Érik**

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des martingales (General theory of processes)

The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in 1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)

Keywords: General filtrations, Strong supermartingales, Snell's envelope, Section theorems

Nature: Original

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XVI: 26, 298-313, LNM 920 (1982)

**DELLACHERIE, Claude**; **LENGLART, Érik**

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (General theory of processes)

This paper is a sequel to 1524. Let $\Theta$ be a*chronology,* i.e., a family of stopping times containing $0$ and $\infty$ and closed under the operations $\land,\lor$---examples are the family of all stopping times, and that of all deterministic stopping times. The general problem discussed is that of defining an optional process $X$ on $[0,\infty]$ such that for each $T\in\Theta$ $X_T$ is a.s. equal to a given r.v. (${\cal F}_T$-measurable). While in 1525 the discussion concerned supermartingales, it is extended here to processes which satisfy a semi-continuity condition from the right

Keywords: Stopping times

Nature: Original

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XVI: 27, 314-318, LNM 920 (1982)

**LENGLART, Érik**

Sur le théorème de la convergence dominée (General theory of processes, Stochastic calculus)

Consider previsible processes $U^n,U$ such that $U^n_T\rightarrow U_T$ in some sense at bounded previsible times $T$. The problem discussed is whether stochastic integrals $\int U^n_s dX_s$ converge (in the same sense) to $\int U_sdX_s$. Under a domination hypothesis, the answer is shown to be positive if the convergence is either weak convergence in $L^1$, or convergence in probability. The existence of the limiting process $U$ is not assumed in the paper; it is proved by a modification of an argument of Mokobodzki for which see 1110

Keywords: Stopping times, Optional processes, Weak convergence, Stochastic integrals

Nature: Original

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XVII: 32, 321-345, LNM 986 (1983)

**LENGLART, Érik**

Désintégration régulière de mesure sans conditions habituelles

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XXI: 19, 276-288, LNM 1247 (1987)

**FOURATI, Sonia**; **LENGLART, Érik**

Tribus homogènes et commutation de projections

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Une caractérisation des processus prévisibles (General theory of processes)

One of the results of this short paper is the following: a bounded optional process $X$ is previsible if and only if, for every martingale $M$ of integrable variation, the Stieltjes integral process $X\sc M$ is a martingale

Keywords: Previsible processes

Nature: Original

Retrieve article from Numdam

XII: 07, 53-56, LNM 649 (1978)

Sur la localisation des intégrales stochastiques (Stochastic calculus)

A mapping $T$ from processes to processes is

Keywords: Girsanov's theorem

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: 05, 49-52, LNM 784 (1980)

Appendice à l'exposé précédent~: inégalités de semimartingales (Martingale theory, Stochastic calculus)

This paper contains several applications of the methods of 1404 to the case of semimartingales instead of martingales

Keywords: Inequalities, Semimartingales

Nature: Original

Retrieve article from Numdam

XIV: 14, 125-127, LNM 784 (1980)

Sur l'inégalité de Métivier-Pellaumail (Stochastic calculus)

A simplified (but still not so simple) proof of the Métivier-Pellaumail inequality

Keywords: Doob's inequality, Métivier-Pellaumail inequality

Nature: New proof of known results

Retrieve article from Numdam

XIV: 49, 500-546, LNM 784 (1980)

Tribus de Meyer et théorie des processus (General theory of processes, Stochastic calculus)

The subject of this paper is the study of the $\sigma$-field on $

Comment: This beautiful paper was generally ignored. If a suggestive name had been used instead of the terminology ``Meyer $\sigma$-field'', its fate might have been different. See 1524 for an interesting application. The work of Fourati (partly unpublished) follows along the same lines, but including time reversal: see 2119

Keywords: Projection theorems, Section theorems

Nature: Original

Retrieve article from Numdam

XV: 24, 320-346, LNM 850 (1981)

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des martingales (General theory of processes)

The optional section theorem implies that an optional process $X$ is completely determined by its values $X_T$ at all stopping times. Conversely, given random variables $X_T$, ${\cal F}_T$-measurable and such that $X_S=X_T$ a.s. on the set $\{S=T\}$, is it possible to ``aggregate'' them into an optional process $X$? This is the elementary form of the general problem discussed in the paper, in the case where the random variables $X_T$ satisfy a supermartingale inequality. The problem solved is more general: the optional $\sigma$-field is replaced by any of the $\sigma$-fields considered in 1449 (including previsible, accessible, etc), and the family of all stopping times is replaced by a suitable family (called a chronology)

Keywords: General filtrations, Strong supermartingales, Snell's envelope, Section theorems

Nature: Original

Retrieve article from Numdam

XVI: 26, 298-313, LNM 920 (1982)

Sur des problèmes de régularisation, de recollement et d'interpolation en théorie des processus (General theory of processes)

This paper is a sequel to 1524. Let $\Theta$ be a

Keywords: Stopping times

Nature: Original

Retrieve article from Numdam

XVI: 27, 314-318, LNM 920 (1982)

Sur le théorème de la convergence dominée (General theory of processes, Stochastic calculus)

Consider previsible processes $U^n,U$ such that $U^n_T\rightarrow U_T$ in some sense at bounded previsible times $T$. The problem discussed is whether stochastic integrals $\int U^n_s dX_s$ converge (in the same sense) to $\int U_sdX_s$. Under a domination hypothesis, the answer is shown to be positive if the convergence is either weak convergence in $L^1$, or convergence in probability. The existence of the limiting process $U$ is not assumed in the paper; it is proved by a modification of an argument of Mokobodzki for which see 1110

Keywords: Stopping times, Optional processes, Weak convergence, Stochastic integrals

Nature: Original

Retrieve article from Numdam

XVII: 32, 321-345, LNM 986 (1983)

Désintégration régulière de mesure sans conditions habituelles

Retrieve article from Numdam

XXI: 19, 276-288, LNM 1247 (1987)

Tribus homogènes et commutation de projections

Retrieve article from Numdam