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3 matches found
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 processes
Nature: Exposition
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XII: 33, 457-467, LNM 649 (1978)
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 1332
Keywords: Optimal stopping, Snell's envelope
Nature: Original
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XIII: 32, 378-384, LNM 721 (1979)
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 theory
Nature: Original
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