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 timesNature: Original
Retrieve article from Numdam
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
906Keywords: Stopping times,
Accessible times,
Previsible timesNature: Original Retrieve article from Numdam
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 processesNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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 theoremsNature: Original Retrieve article from Numdam
VII: 18, 180-197, LNM 321 (1973)
MEYER, Paul-André
Résultats d'Azéma en théorie générale des processus (
General theory of processes)
This paper presents several results from a paper of Azéma (
Invent. Math.,
18, 1972) which have become (in a slightly extended version) standard tools in the general theory of processes. The problem is that of ``localizing'' a time $L$ which is not a stopping time. With $L$ are associated the supermartingale $c^L_t=P\{L>t|{\cal F}_t\}$ and the previsible increasing processes $p^L$ which generates it (and is the dual previsible projection of the unit mass on the graph of $L$). Then the left support of $dp^L$ is the smallest left-closed previsible set containing the graph of $L$, while the set $\{c^L_-=1\}$ is the greatest previsible set to the left of $L$. Other useful results are the following: given a progressive process $X$, the process $\limsup_{s\rightarrow t} X_s$ is optional, previsible if $s<t$ is added, and a few similar results
Comment: These results have been included (with their optional counterpart, whose interest was discovered later) in Dellacherie-Meyer,
Probabilités et Potentiel, Vol. E, Chapter XX
12--17
Keywords: Optimal stopping,
Previsible processesNature: Exposition Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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 formulaNature: Original Retrieve article from Numdam
XI: 06, 51-58, LNM 581 (1977)
DUDLEY, Richard M.;
GUTMANN, Sam
Stopping times with given laws (
General theory of processes)
Given a filtration ${\cal A}_t$ such that for all $t>0$ the law $P$ restricted to ${\cal A}_t$ is non-atomic, then for every law $\mu$ on the half-line there exists a stopping time with law $\mu$. The proof uses an interesting measure theoretic lemma on decreasing sequences of non-atomic $\sigma$-fields
Comment: It follows that the Brownian filtration contains a process whose law is that of a Poisson process (though of course it is not a Poisson process
of the Brownian filtration )
Keywords: StoppingNature: Original Retrieve article from Numdam
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
1308Keywords: Totally inaccessible stopping timesNature: Original Retrieve article from Numdam
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 timesNature: Exposition,
Original additions Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
XII: 33, 457-467, LNM 649 (1978)
MAINGUENEAU, Marie Anne
Temps d'arrêt optimaux et théorie générale (
General theory of processes)
This is a general discussion of optimal stopping in continuous time. Fairly advanced tools like strong supermartingales, Mertens' decomposition are used
Comment: The subject is taken up in
1332Keywords: Optimal stopping,
Snell's envelopeNature: Original Retrieve article from Numdam
XIII: 14, 174-198, LNM 721 (1979)
CAIROLI, Renzo;
GABRIEL, Jean-Pierre
Arrêt de certaines suites multiples de variables aléatoires indépendantes (
Several parameter processes,
Independence)
Let $(X_n)$ be independent, identically distributed random variables. It is known that $X_T/T\in L^1$ for all stopping times $T$ (or the same with $S_n=X_1+...+X_n$ replacing $X_n$) if and only if $X\in L\log L$. The problem is to extend this to several dimensions, $
N^d$ ($d>1$) replacing $
N$. Then a stopping time $T$ becomes a stopping point, of which two definitions can be given (the past at time $n$ being defined either as the past rectangle, or the complement of the future rectangle), and $|T|$ being defined as the product of the coordinates). The appropriate space then is $L\log L$ or $L\log^d L$ depending on the kind of stopping times involved. Also the integrability of the supremum of the processes along random increasing paths is considered
Keywords: Stopping points,
Random increasing pathsNature: Original Retrieve article from Numdam
XIII: 32, 378-384, LNM 721 (1979)
SZPIRGLAS, Jacques;
MAZZIOTTO, Gérald
Théorème de séparation dans le problème d'arrêt optimal (
General theory of processes)
Let $({\cal G}_t)$ be an enlargement of a filtration $({\cal F}_t)$ with the property that for every $t$, if $X$ is ${\cal G}_t$-measurable, then $E[X\,|\,{\cal F}_t]=E[X\,|\,{\cal F}_\infty]$. Then if $(X_t)$ is a ${\cal F}$-optional process, its Snell envelope is the same in both filtrations. Applications are given to filtering theory
Keywords: Optimal stopping,
Snell's envelope,
Filtering theoryNature: Original Retrieve article from Numdam
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 processesNature: Original Retrieve article from Numdam
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 timesNature: Original Retrieve article from Numdam
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
1110Keywords: Stopping times,
Optional processes,
Weak convergence,
Stochastic integralsNature: Original Retrieve article from Numdam
XLV: 11, 301-304, LNM 2078 (2013)
SOKOL, Alexander
An Elementary Proof that the First Hitting Time of an Open Set by a Jump Process is a Stopping Time (
Theory of processes)
Keywords: Stopping time,
Jump process,
First hitting timeNature: Original
