Quick search | Browse volumes | |

V: 30, 311-341, LNM 191 (1971)

**WATANABE, Takesi**

On balayées of excessive measures and functions with respect to resolvents (Potential theory)

A general study of balayage of excessive measures as dual to réduite of excessive functions, first for a single kernel, then for a resolvent on a measurable space, and finally for a standard process

Comment: See Kunita and T. Watanabe,*Ill. J. Math.*, **9**, 1965. For the modern theory of balayage of measures (using Kuznetsov's processes) see Getoor, *Excessive Measures,* 1990, Chapter 4

Keywords: Excessive measures, Balayage

Nature: Original

Retrieve article from Numdam

VI: 12, 130-150, LNM 258 (1972)

**MEYER, Paul-André**

Le schéma de remplissage en temps continu, d'après H. Rost (Ergodic theory, Potential theory)

The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as 523. Here following Rost himself (*Invent. Math.,* **14**, 1971) the construction is extended to continuous time Markov processes. In the transient case, the results are translated in potential-theoretic language, and proved using techniques due to Mokobodzki. Then the general case follows from this result applied to a space-time extension of the semi-group

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Filling scheme, Balayage of measures, Skorohod imbedding

Nature: Exposition, Original additions

Retrieve article from Numdam

VII: 16, 155-171, LNM 321 (1973)

**MEYER, Paul-André**; **TRAKI, Mohammed**

Réduites et jeux de hasard (Potential theory)

This paper arose from an attempt (by the second author) to rewrite the results of Dubins-Savage*How to Gamble if you Must * in the language of standard (countably additive) measure theory, using the methods of descriptive set theory (analytic sets, section theorems, etc). The attempt is successful, since all general theorems can be proved in this set-up. More recent results in the same line, due to Strauch and Sudderth, are extended too. An appendix includes useful comments by Mokobodzki on the case of a gambling house consisting of a single kernel (discrete potential theory)

Comment: This material is reworked in Dellacherie-Meyer,*Probabilités et Potentiel,* Vol. C, Chapter X

Keywords: Balayage, Gambling house, Réduite, Optimal strategy

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

XIII: 40, 472-477, LNM 721 (1979)

**STRICKER, Christophe**

Semimartingales et valeur absolue (General theory of processes)

For the general notation, see 1338. A result of Yoeurp that absolute values preserves quasimartingales is extended: convex functions satisfying a Lipschitz condition operate on quasimartingales. For $p\ge1$, $X\in H^p$ implies $|X|^p\in H^1$. Then it is shown that for a continuous adapted process $X$, it is equivalent to say that $X$ and $|X|$ are quasimartingales (or semimartingales). Then comes a result related to the main problem of this series: with the general notations above, if $X$ is assumed to be a quasimartingale such that $X_{D_t}=0$ for all $t$, if the process $Z$ is progressive and bounded, then the process $Z_{g_t}X_t$ is a quasimartingale

Comment: A complement is given in the next paper 1341. See also 1351

Keywords: Balayage, Quasimartingales

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

XIII: 51, 610-610, LNM 721 (1979)

**STRICKER, Christophe**

Encore une remarque sur la ``formule de balayage'' (General theory of processes)

A slight extension of 1341

Keywords: Balayage

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

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

On balayées of excessive measures and functions with respect to resolvents (Potential theory)

A general study of balayage of excessive measures as dual to réduite of excessive functions, first for a single kernel, then for a resolvent on a measurable space, and finally for a standard process

Comment: See Kunita and T. Watanabe,

Keywords: Excessive measures, Balayage

Nature: Original

Retrieve article from Numdam

VI: 12, 130-150, LNM 258 (1972)

Le schéma de remplissage en temps continu, d'après H. Rost (Ergodic theory, Potential theory)

The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as 523. Here following Rost himself (

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Filling scheme, Balayage of measures, Skorohod imbedding

Nature: Exposition, Original additions

Retrieve article from Numdam

VII: 16, 155-171, LNM 321 (1973)

Réduites et jeux de hasard (Potential theory)

This paper arose from an attempt (by the second author) to rewrite the results of Dubins-Savage

Comment: This material is reworked in Dellacherie-Meyer,

Keywords: Balayage, Gambling house, Réduite, Optimal strategy

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: 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: 40, 472-477, LNM 721 (1979)

Semimartingales et valeur absolue (General theory of processes)

For the general notation, see 1338. A result of Yoeurp that absolute values preserves quasimartingales is extended: convex functions satisfying a Lipschitz condition operate on quasimartingales. For $p\ge1$, $X\in H^p$ implies $|X|^p\in H^1$. Then it is shown that for a continuous adapted process $X$, it is equivalent to say that $X$ and $|X|$ are quasimartingales (or semimartingales). Then comes a result related to the main problem of this series: with the general notations above, if $X$ is assumed to be a quasimartingale such that $X_{D_t}=0$ for all $t$, if the process $Z$ is progressive and bounded, then the process $Z_{g_t}X_t$ is a quasimartingale

Comment: A complement is given in the next paper 1341. See also 1351

Keywords: Balayage, Quasimartingales

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: 51, 610-610, LNM 721 (1979)

Encore une remarque sur la ``formule de balayage'' (General theory of processes)

A slight extension of 1341

Keywords: Balayage

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