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III: 10, 152-154, LNM 88 (1969)

**MEYER, Paul-André**

Un résultat élémentaire sur les temps d'arrêt (General theory of processes)

This useful result asserts that a stopping time is accessible if and only if its graph is contained in a countable union of graphs of previsible stopping times

Comment: Before this was noticed, accessible stopping times were considered important. After this remark, previsible stopping times came to the forefront

Keywords: Stopping times, Accessible times, Previsible times

Nature: Original

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IV: 05, 60-70, LNM 124 (1970)

**DELLACHERIE, Claude**

Un exemple de la théorie générale des processus (General theory of processes)

In the case of the smallest filtration for which a given random variable is a stopping time, all the computations of the general theory can be performed explicitly

Comment: This example has become classical. See for example Dellacherie-Meyer,*Probabilités et Potentiel,* Chap IV. On the other hand, it can be extended to deal with (unmarked) point processes: see Chou-Meyer 906

Keywords: Stopping times, Accessible times, Previsible times

Nature: Original

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VI: 14, 159-163, LNM 258 (1972)

**MEYER, Paul-André**

Temps d'arrêt algébriquement prévisibles (General theory of processes)

The main results concern the natural filtration of a right continuous process taking values in a Polish spaces, and defined on a Blackwell space $\Omega$. Conditions are given on a process or a random variable on $\Omega$ which insure that it will be previsible or optional under any probability law on $\Omega$

Comment: The subject has been kept alive by Azéma, who used similar techniques in several papers

Keywords: Stopping times, Previsible processes

Nature: Original

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VII: 04, 36-37, LNM 321 (1973)

**DELLACHERIE, Claude**

Temps d'arrêt totalement inaccessibles (General theory of processes)

Given an accessible random set $H$ and a totally inaccessible stopping time $T$, whenever $T(\omega)\in H(\omega)$ then $T(\omega)$ is a condensation point of $H(\omega)$ on the left, i.e., there are uncountably many points of $H$ arbitrarily close to $T(\omega)$ on the left

Keywords: Stopping times, Accessible sets, Totally inaccessible stopping times

Nature: Original

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VII: 05, 38-47, LNM 321 (1973)

**DELLACHERIE, Claude**

Sur les théorèmes fondamentaux de la théorie générale des processus (General theory of processes)

This paper reconstructs the general theory of processes starting from a suitable family ${\cal V }$ of stopping times, and the $\sigma$-field generated by stochastic intervals $[S,T[$ with $S,T\in{\cal V }$, $S\le T$. Section and projection theorems are proved

Comment: The idea of this paper has proved fruitful. See for instance Lenglart, 1449; Le Jan,*Z. für W-Theorie * **44**, 1978

Keywords: Stopping times, Section theorems

Nature: Original

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IX: 16, 373-389, LNM 465 (1975)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

Ensembles analytiques et temps d'arrêt (Descriptive set theory)

This is a sequel to the preceding paper 915. Instead of using the language of trees to prove the second separation theorem, a language more familiar to probabilists is used, in which the space of stopping times on $**N**^**N**$ is given a compact metric topology and the space of non-finite stopping times appears as the universal analytic, non-Borel set, from which all analytic sets can be constructed. Many proofs become very natural in this language

Comment: See also the next paper 917, the set of lectures by Dellacherie in C.A. Rogers,*Analytic Sets,* Academic Press 1981, and chapter XXIV of Dellacherie-Meyer, *Probabilités et potentiel *

Keywords: Second separation theorem, Stopping times

Nature: Original

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IX: 36, 555-555, LNM 465 (1975)

**MEYER, Paul-André**

Une remarque sur les processus de Markov (Markov processes)

It is shown that, under a fixed measure $**P**^{\mu}$, the optional processes and times relative to the uncompleted filtrations $({\cal F}_{t+}^{\circ})$ and $({\cal F}_{t}^{\circ})$ are undistinguishable from each other

Comment: No applications are known

Keywords: Stopping times

Nature: Original

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X: 15, 235-239, LNM 511 (1976)

**WILLIAMS, David**

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see*Ann. Prob.* **3**, 1975. For modern references, we should ask Yor

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

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XII: 06, 51-52, LNM 649 (1978)

**GARCIA, M.**; **MAILLARD, P.**; **PELTRAUT, Y.**

Une martingale de saut multiplicatif donné (Martingale theory)

Given a totally inaccessible stopping time $T$, it is shown how to construct a strictly positive martingale $M$ with $M_0=1$, such that its only jump occurs at time $T$ and $M_T/M_{T-}=K$, a strictly positive constant

Comment: See also 1308

Keywords: Totally inaccessible stopping times

Nature: Original

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XII: 28, 411-423, LNM 649 (1978)

**MEYER, Paul-André**

Convergence faible et compacité des temps d'arrêt, d'après Baxter et Chacón (General theory of processes)

Baxter and Chacón (*Zeit. für W-theorie,* 40, 1977) introduced a topology on the sets of ``fuzzy'' times and of fuzzy stopping times which turn these sets into compact metrizable spaces---a fuzzy r.v. $T$ is a right continuous decreasing process $M_t$ with $M_{0-}=1$, $M_t(\omega)$ being interpreted for each $\omega$ as the distribution function $P_{\omega}\{T>t\}$. When this process is adapted the fuzzy r.v. is a fuzzy stopping time. A number of properties of this topology are investigated

Comment: See 1536 for an extension to Polish spaces

Keywords: Stopping times, Fuzzy stopping times

Nature: Exposition, Original additions

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XII: 29, 424-424, LNM 649 (1978)

**DELLACHERIE, Claude**

Convergence en probabilité et topologie de Baxter-Chacón (General theory of processes)

It is shown that on the set of ordinary stopping times, the Baxter-Chacón topology is simply convergence in probability

Keywords: Stopping times

Nature: Original

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XII: 30, 425-427, LNM 649 (1978)

**DELLACHERIE, Claude**; **MEYER, Paul-André**

Construction d'un processus prévisible ayant une valeur donnée en un temps d'arrêt (General theory of processes)

Let $T$ be a stopping time, $X$ be an integrable r.v., and put $A_t=I_{\{t\ge T\}}$ and $B_t=XA_t$. Then the previsible compensator $(\tilde B_t)$ has a previsible density $Z_t$ with respect to $(\tilde A_t)$, whose value $Z_T$ at time $T$ is $E[X\,|\,{\cal F}_{T-}]$, and in particular if $X$ is ${\cal F}_T$-measurable it is equal to $X$

Keywords: Stopping times

Nature: Original

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XVI: 17, 213-218, LNM 920 (1982)

**FALKNER, Neil**; **STRICKER, Christophe**; **YOR, Marc**

Temps d'arrêt riches et applications (General theory of processes)

This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see 1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper 1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$

Keywords: Stopping times, Local times, Semimartingales, Previsible processes

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

Retrieve article from Numdam

Un résultat élémentaire sur les temps d'arrêt (General theory of processes)

This useful result asserts that a stopping time is accessible if and only if its graph is contained in a countable union of graphs of previsible stopping times

Comment: Before this was noticed, accessible stopping times were considered important. After this remark, previsible stopping times came to the forefront

Keywords: Stopping times, Accessible times, Previsible times

Nature: Original

Retrieve article from Numdam

IV: 05, 60-70, LNM 124 (1970)

Un exemple de la théorie générale des processus (General theory of processes)

In the case of the smallest filtration for which a given random variable is a stopping time, all the computations of the general theory can be performed explicitly

Comment: This example has become classical. See for example Dellacherie-Meyer,

Keywords: Stopping times, Accessible times, Previsible times

Nature: Original

Retrieve article from Numdam

VI: 14, 159-163, LNM 258 (1972)

Temps d'arrêt algébriquement prévisibles (General theory of processes)

The main results concern the natural filtration of a right continuous process taking values in a Polish spaces, and defined on a Blackwell space $\Omega$. Conditions are given on a process or a random variable on $\Omega$ which insure that it will be previsible or optional under any probability law on $\Omega$

Comment: The subject has been kept alive by Azéma, who used similar techniques in several papers

Keywords: Stopping times, Previsible processes

Nature: Original

Retrieve article from Numdam

VII: 04, 36-37, LNM 321 (1973)

Temps d'arrêt totalement inaccessibles (General theory of processes)

Given an accessible random set $H$ and a totally inaccessible stopping time $T$, whenever $T(\omega)\in H(\omega)$ then $T(\omega)$ is a condensation point of $H(\omega)$ on the left, i.e., there are uncountably many points of $H$ arbitrarily close to $T(\omega)$ on the left

Keywords: Stopping times, Accessible sets, Totally inaccessible stopping times

Nature: Original

Retrieve article from Numdam

VII: 05, 38-47, LNM 321 (1973)

Sur les théorèmes fondamentaux de la théorie générale des processus (General theory of processes)

This paper reconstructs the general theory of processes starting from a suitable family ${\cal V }$ of stopping times, and the $\sigma$-field generated by stochastic intervals $[S,T[$ with $S,T\in{\cal V }$, $S\le T$. Section and projection theorems are proved

Comment: The idea of this paper has proved fruitful. See for instance Lenglart, 1449; Le Jan,

Keywords: Stopping times, Section theorems

Nature: Original

Retrieve article from Numdam

IX: 16, 373-389, LNM 465 (1975)

Ensembles analytiques et temps d'arrêt (Descriptive set theory)

This is a sequel to the preceding paper 915. Instead of using the language of trees to prove the second separation theorem, a language more familiar to probabilists is used, in which the space of stopping times on $

Comment: See also the next paper 917, the set of lectures by Dellacherie in C.A. Rogers,

Keywords: Second separation theorem, Stopping times

Nature: Original

Retrieve article from Numdam

IX: 36, 555-555, LNM 465 (1975)

Une remarque sur les processus de Markov (Markov processes)

It is shown that, under a fixed measure $

Comment: No applications are known

Keywords: Stopping times

Nature: Original

Retrieve article from Numdam

X: 15, 235-239, LNM 511 (1976)

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

Retrieve article from Numdam

XII: 06, 51-52, LNM 649 (1978)

Une martingale de saut multiplicatif donné (Martingale theory)

Given a totally inaccessible stopping time $T$, it is shown how to construct a strictly positive martingale $M$ with $M_0=1$, such that its only jump occurs at time $T$ and $M_T/M_{T-}=K$, a strictly positive constant

Comment: See also 1308

Keywords: Totally inaccessible stopping times

Nature: Original

Retrieve article from Numdam

XII: 28, 411-423, LNM 649 (1978)

Convergence faible et compacité des temps d'arrêt, d'après Baxter et Chacón (General theory of processes)

Baxter and Chacón (

Comment: See 1536 for an extension to Polish spaces

Keywords: Stopping times, Fuzzy stopping times

Nature: Exposition, Original additions

Retrieve article from Numdam

XII: 29, 424-424, LNM 649 (1978)

Convergence en probabilité et topologie de Baxter-Chacón (General theory of processes)

It is shown that on the set of ordinary stopping times, the Baxter-Chacón topology is simply convergence in probability

Keywords: Stopping times

Nature: Original

Retrieve article from Numdam

XII: 30, 425-427, LNM 649 (1978)

Construction d'un processus prévisible ayant une valeur donnée en un temps d'arrêt (General theory of processes)

Let $T$ be a stopping time, $X$ be an integrable r.v., and put $A_t=I_{\{t\ge T\}}$ and $B_t=XA_t$. Then the previsible compensator $(\tilde B_t)$ has a previsible density $Z_t$ with respect to $(\tilde A_t)$, whose value $Z_T$ at time $T$ is $E[X\,|\,{\cal F}_{T-}]$, and in particular if $X$ is ${\cal F}_T$-measurable it is equal to $X$

Keywords: Stopping times

Nature: Original

Retrieve article from Numdam

XVI: 17, 213-218, LNM 920 (1982)

Temps d'arrêt riches et applications (General theory of processes)

This paper starts from the existence of increasing left-continuous processes $(A_t)$ which generate the previsible $\sigma$-field, i.e., every previsible process can be represented as $f(X_t)$ for some Borel function $f$ (see 1123), to prove the existence (discovered by the first named author) of ``rich'' stopping times $T$, i.e., previsible stopping times which encode the whole past up to time $T$: $\sigma(T)={\cal F}_{T-}$ (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper 1616) and a stopping time $T$ for Brownian motion such that $L^a_T$ is not a semimartingale in its space variable $a$

Keywords: Stopping times, Local times, Semimartingales, Previsible processes

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