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VII: 11, 95-117, LNM 321 (1973)

**EL KAROUI, Nicole**; **REINHARD, Hervé**

Processus de diffusion dans ${\bf R}^n$ (Diffusion theory)

This paper concerns diffusions (without boundaries) whose generators have Borel bounded coefficients. It consists of two parts. The first one is devoted to the equivalence between the existence and uniqueness of the diffusion semigroup and the uniqueness in law of the solution of the corresponding Ito stochastic differential equation. This allows the authors to use in the elliptic case the deep results of Krylov on s.d.e.'s. The second part concerns mostly the Lipschitz case, and discusses several properties of the diffusion process in itself: the representation of additive functional martingales; the relations between the number of martingales necessary for the representation and the rank of the generator (locally); the existence of a dual diffusion; the support and absolute continuity properties of the semi-group

Comment: This paper is in part an improved version of a paper on degenerate diffusions by Bonami, El-Karoui, Reinhard and Roynette (*Ann. Inst. H. Poincaré,* **7**, 1971)

Keywords: Construction of diffusions, Diffusions with measurable coefficients, Degenerate diffusions

Nature: Original

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IX: 35, 534-554, LNM 465 (1975)

**EL KAROUI, Nicole**

Processus de réflexion dans ${\bf R}^n$ (Diffusion theory)

In the line of the seminar on diffusions 419 this talk presents the theory of diffusions in a half space with continuous coefficients and a boundary condition on the boundary hyperplane involving a reflexion part, but more general than the pure reflexion case considered by Stroock-Varadhan (*Comm. Pure Appl. Math.*, **24**, 1971). The point of view is that of martingale problems

Comment: This talk is a late publication of work done by the author in 1971

Keywords: Boundary reflection, Local times

Nature: Original

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XI: 08, 65-78, LNM 581 (1977)

**EL KAROUI, Nicole**; **MEYER, Paul-André**

Les changements de temps en théorie générale des processus (General theory of processes)

Given a filtration $({\cal F}_t)$ and a continuous adapted increasing process $(C_t)$, consider its right inverse $(j_t)$ and left inverse $(i_t)$, and the time-changed filtration $\overline{\cal F}_t={\cal F}_{j_t}$. The problem is to study the relation between optional/previsible processes of the time-changed filtration and time-changed optional/previsible processes of the original filtration, to see how the projections or dual projections are related, etc. The results are satisfactory, and require a lot of care

Comment: This paper was originally an exposition by the second author of an unpublished paper of the first author, and many ``I''s remained in spite of the final joint autorship. See the next paper 1109 for the discontinuous case

Keywords: Changes of time

Nature: Original

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XI: 09, 79-108, LNM 581 (1977)

**EL KAROUI, Nicole**; **WEIDENFELD, Gérard**

Théorie générale et changement de temps (General theory of processes)

The results of the preceding paper 1108 are extended to arbitrary changes of times, i.e., without the continuity assumption on the increasing process. They require even more care

Comment: Unfortunately, the material presentation of this paper is rather poor. For related results, see 1333

Keywords: Changes of time

Nature: Original

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XIII: 38, 443-452, LNM 721 (1979)

**EL KAROUI, Nicole**

Temps local et balayage des semimartingales (General theory of processes)

This paper is the first one in a series of reports on the balayage of semimartingales, and the following description is common to all of them. \par Let $H$ be a right-closed optional set, and let $g_t=\sup\{s<t, s\in H\}$ and $D_t=\inf\{s>t,s\in H\}$. Put $L=g_{\infty}$. Let also $G$ be the set of all left-endpoints of intervals contiguous to $H$, i.e., of all points $g_t$ for $t\notin H$. For simplicity we assume here that $D_0=0$ and that $H=\{X=0\}$, where $X$ is a semimartingale with decomposition $X=M+V$, though for a few results (including the balayage formula itself) it is sufficient that $X=0$ on $H$. \par One of the starting points of this paper is the*balayage formula * (see Azéma-Yor, introduction to *Temps Locaux *, *Astérisque *, **52-53**): if $Z$ is a locally bounded previsible process, then $$Z_{g_t}X_t=\int_0^t Z_{g_s}dX_s$$ and therefore $Y_t=Z_{g_t}X_t$ is a semimartingale. The main problem of the series of reports is: what can be said if $Z$ is not previsible, but optional, or even progressive?\par This particular paper is devoted to the study of the non-adapted process $$K_t=\sum_{g\in G,g\le t } (M_{D_g}-M_g)$$ which turns out to have finite variation

Comment: This paper is completed by 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

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XIII: 57, 634-641, LNM 721 (1979)

**EL KAROUI, Nicole**

A propos de la formule d'Azéma-Yor (General theory of processes)

For the problem and notation, see the review of 1340. The problem is completely solved here, the process $Z_{g_t}X_t$ being represented as the sum of $\int_0^t Z_{g_s}dX_s$ interpreted in a generalized sense ($Z$ being progressive!) and a remainder which can be explicitly written (using optional dual projections of non-adapted processes)

Comment: This paper ends happily the whole series of papers on balayage in this volume

Keywords: Balayage, Balayage formula

Nature: Original

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XVI: 36, 400-408, LNM 920 (1982)

**EL KAROUI, Nicole**

Une propriété de domination de l'enveloppe de Snell des semimartingales fortes

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XXII: 39, 508-541, LNM 1321 (1988)

**EL KAROUI, Nicole**; **JEANBLANC, Monique**

Contrôle de processus de Markov

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XXIII: 34, 405-420, LNM 1372 (1989)

**EL KAROUI, Nicole**; **KARATZAS, Ioannis**

Integration of the optimal risk in a stopping problem with absorption

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Processus de diffusion dans ${\bf R}^n$ (Diffusion theory)

This paper concerns diffusions (without boundaries) whose generators have Borel bounded coefficients. It consists of two parts. The first one is devoted to the equivalence between the existence and uniqueness of the diffusion semigroup and the uniqueness in law of the solution of the corresponding Ito stochastic differential equation. This allows the authors to use in the elliptic case the deep results of Krylov on s.d.e.'s. The second part concerns mostly the Lipschitz case, and discusses several properties of the diffusion process in itself: the representation of additive functional martingales; the relations between the number of martingales necessary for the representation and the rank of the generator (locally); the existence of a dual diffusion; the support and absolute continuity properties of the semi-group

Comment: This paper is in part an improved version of a paper on degenerate diffusions by Bonami, El-Karoui, Reinhard and Roynette (

Keywords: Construction of diffusions, Diffusions with measurable coefficients, Degenerate diffusions

Nature: Original

Retrieve article from Numdam

IX: 35, 534-554, LNM 465 (1975)

Processus de réflexion dans ${\bf R}^n$ (Diffusion theory)

In the line of the seminar on diffusions 419 this talk presents the theory of diffusions in a half space with continuous coefficients and a boundary condition on the boundary hyperplane involving a reflexion part, but more general than the pure reflexion case considered by Stroock-Varadhan (

Comment: This talk is a late publication of work done by the author in 1971

Keywords: Boundary reflection, Local times

Nature: Original

Retrieve article from Numdam

XI: 08, 65-78, LNM 581 (1977)

Les changements de temps en théorie générale des processus (General theory of processes)

Given a filtration $({\cal F}_t)$ and a continuous adapted increasing process $(C_t)$, consider its right inverse $(j_t)$ and left inverse $(i_t)$, and the time-changed filtration $\overline{\cal F}_t={\cal F}_{j_t}$. The problem is to study the relation between optional/previsible processes of the time-changed filtration and time-changed optional/previsible processes of the original filtration, to see how the projections or dual projections are related, etc. The results are satisfactory, and require a lot of care

Comment: This paper was originally an exposition by the second author of an unpublished paper of the first author, and many ``I''s remained in spite of the final joint autorship. See the next paper 1109 for the discontinuous case

Keywords: Changes of time

Nature: Original

Retrieve article from Numdam

XI: 09, 79-108, LNM 581 (1977)

Théorie générale et changement de temps (General theory of processes)

The results of the preceding paper 1108 are extended to arbitrary changes of times, i.e., without the continuity assumption on the increasing process. They require even more care

Comment: Unfortunately, the material presentation of this paper is rather poor. For related results, see 1333

Keywords: Changes of time

Nature: Original

Retrieve article from Numdam

XIII: 38, 443-452, LNM 721 (1979)

Temps local et balayage des semimartingales (General theory of processes)

This paper is the first one in a series of reports on the balayage of semimartingales, and the following description is common to all of them. \par Let $H$ be a right-closed optional set, and let $g_t=\sup\{s<t, s\in H\}$ and $D_t=\inf\{s>t,s\in H\}$. Put $L=g_{\infty}$. Let also $G$ be the set of all left-endpoints of intervals contiguous to $H$, i.e., of all points $g_t$ for $t\notin H$. For simplicity we assume here that $D_0=0$ and that $H=\{X=0\}$, where $X$ is a semimartingale with decomposition $X=M+V$, though for a few results (including the balayage formula itself) it is sufficient that $X=0$ on $H$. \par One of the starting points of this paper is the

Comment: This paper is completed by 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 57, 634-641, LNM 721 (1979)

A propos de la formule d'Azéma-Yor (General theory of processes)

For the problem and notation, see the review of 1340. The problem is completely solved here, the process $Z_{g_t}X_t$ being represented as the sum of $\int_0^t Z_{g_s}dX_s$ interpreted in a generalized sense ($Z$ being progressive!) and a remainder which can be explicitly written (using optional dual projections of non-adapted processes)

Comment: This paper ends happily the whole series of papers on balayage in this volume

Keywords: Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XVI: 36, 400-408, LNM 920 (1982)

Une propriété de domination de l'enveloppe de Snell des semimartingales fortes

Retrieve article from Numdam

XXII: 39, 508-541, LNM 1321 (1988)

Contrôle de processus de Markov

Retrieve article from Numdam

XXIII: 34, 405-420, LNM 1372 (1989)

Integration of the optimal risk in a stopping problem with absorption

Retrieve article from Numdam