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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: 39, 453-471, LNM 721 (1979)

**YOR, Marc**

Sur le balayage des semi-martingales continues (General theory of processes)

For the general notation, see 1338. This paper is independent from the preceding one 1338, and some overlap occurs. The balayage formula is extended to processes $Z$ which are not locally bounded, and the local time of the semimartingale $Y$ is computed. The class of continuous semimartingales $X$ with canonical decomposition $X=M+V$ such that $dV$ is carried by $H=\{X=0\}$ is introduced and studied. It turns out to be an important class, closely related to ``relative martingales'' (Azéma, Meyer and Yor 2623). A number of results are given, too technical to be stated here. Stopping previsible, optional and progressive processes at the last exit time $L$ from $H$ leads to three $\sigma$-fields, ${\cal F}_L^p$, ${\cal F}_L^o$, ${\cal F}_L^{\pi}$, and it was considered surprising that the last two could be different (see 1240). Here it is shown that if $X$ is a continuous uniformly integrable martingale with $X_0=0$, $E[X_{\infty}|{\cal F}_L^o]=0\neq E[X_{\infty}|{\cal F}_L^{\pi}]$

Comment: See 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

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XIII: 41, 478-487, LNM 721 (1979)

**MEYER, Paul-André**; **STRICKER, Christophe**; **YOR, Marc**

Sur une formule de la théorie du balayage (General theory of processes)

For the notation, see the review of 1340. It is shown here that under the same hypotheses, the semimartingale $Z_{g_t}X_t$ is a sum of three terms: the stochastic integral $\int_0^t \zeta_s dX_s$, where $\zeta$ is the previsible projection of $Z$, an explicit sum of jumps involving $Z-\zeta$, and a mysterious continuous process with finite variation $(R_t)$ such that $dR_t$ is carried by $H$, equal to $0$ if $Z$ was optional

Comment: See 1351, 1357

Keywords: Balayage, Balayage formula

Nature: Original

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XIII: 55, 624-624, LNM 721 (1979)

**YOR, Marc**

Un exemple de J. Pitman (General theory of processes)

The balayage formula allows the construction of many martingales vanishing on the zeros of a given continuous martingale $X$, namely martingales of the form $Z_{g_t}X_t$ where $Z$ is previsible. Taking $X$ to be Brownian motion, an example is given of a martingale vanishing on its zeros which is not of the above form

Comment: The general problem of finding all martingales which vanish on the zeros of a given continuous martingale is discussed by Azéma and Yor in 2622

Keywords: Balayage, Balayage formula

Nature: Exposition

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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

Retrieve article from Numdam

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: 39, 453-471, LNM 721 (1979)

Sur le balayage des semi-martingales continues (General theory of processes)

For the general notation, see 1338. This paper is independent from the preceding one 1338, and some overlap occurs. The balayage formula is extended to processes $Z$ which are not locally bounded, and the local time of the semimartingale $Y$ is computed. The class of continuous semimartingales $X$ with canonical decomposition $X=M+V$ such that $dV$ is carried by $H=\{X=0\}$ is introduced and studied. It turns out to be an important class, closely related to ``relative martingales'' (Azéma, Meyer and Yor 2623). A number of results are given, too technical to be stated here. Stopping previsible, optional and progressive processes at the last exit time $L$ from $H$ leads to three $\sigma$-fields, ${\cal F}_L^p$, ${\cal F}_L^o$, ${\cal F}_L^{\pi}$, and it was considered surprising that the last two could be different (see 1240). Here it is shown that if $X$ is a continuous uniformly integrable martingale with $X_0=0$, $E[X_{\infty}|{\cal F}_L^o]=0\neq E[X_{\infty}|{\cal F}_L^{\pi}]$

Comment: See 1357

Keywords: Local times, Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 41, 478-487, LNM 721 (1979)

Sur une formule de la théorie du balayage (General theory of processes)

For the notation, see the review of 1340. It is shown here that under the same hypotheses, the semimartingale $Z_{g_t}X_t$ is a sum of three terms: the stochastic integral $\int_0^t \zeta_s dX_s$, where $\zeta$ is the previsible projection of $Z$, an explicit sum of jumps involving $Z-\zeta$, and a mysterious continuous process with finite variation $(R_t)$ such that $dR_t$ is carried by $H$, equal to $0$ if $Z$ was optional

Comment: See 1351, 1357

Keywords: Balayage, Balayage formula

Nature: Original

Retrieve article from Numdam

XIII: 55, 624-624, LNM 721 (1979)

Un exemple de J. Pitman (General theory of processes)

The balayage formula allows the construction of many martingales vanishing on the zeros of a given continuous martingale $X$, namely martingales of the form $Z_{g_t}X_t$ where $Z$ is previsible. Taking $X$ to be Brownian motion, an example is given of a martingale vanishing on its zeros which is not of the above form

Comment: The general problem of finding all martingales which vanish on the zeros of a given continuous martingale is discussed by Azéma and Yor in 2622

Keywords: Balayage, Balayage formula

Nature: Exposition

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