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10 matches found
XII: 08, 57-60, LNM 649 (1978)
MEYER, Paul-André
Sur un théorème de J. Jacod (General theory of processes)
Consider a given process $X$ adapted to a given filtration $({\cal F}_t)$. The set of laws of semimartingales consists of those laws $P$ under which $X$ is a semimartingale with respect to $({\cal F}_t)$ suitably completed. Jacod proved that the set of laws of semimartingales is convex. This is extended here to countable convex combinations, and to integrals
Comment: This easy paper has some historical interest, as it raised the problem of initial enlargement of a filtration
Keywords: Semimartingales, Enlargement of filtrations, Laws of semimartingales
Nature: Original
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XII: 09, 61-69, LNM 649 (1978)
YOR, Marc
Grossissement d'une filtration et semi-martingales~: théorèmes généraux (General theory of processes)
Given a filtration $({\cal F}_t)$ and a positive random variable $L$, the so-called progressively enlarged filtration is the smallest one $({\cal G}_t)$ containing $({\cal F}_t)$, and for which $L$ is a stopping time. The enlargement problem consists in describing the semimartingales $X$ of ${\cal F}$ which remain semimartingales in ${\cal G}$, and in computing their semimartingale characteristics. In this paper, it is proved that $X_tI_{\{t< L\}}$ is a semimartingale in full generality, and that $X_tI_{\{t\ge L\}}$ is a semimartingale whenever $L$ is honest for $\cal F$, i.e., is the end of an $\cal F$-optional set
Comment: This result was independently discovered by Barlow, Zeit. für W-theorie, 44, 1978, which also has a huge intersection with 1211. Complements are given in 1210, and an explicit decomposition formula for semimartingales in 1211
Keywords: Enlargement of filtrations, Honest times
Nature: Original
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XII: 10, 70-77, LNM 649 (1978)
DELLACHERIE, Claude; MEYER, Paul-André
A propos du travail de Yor sur le grossissement des tribus (General theory of processes)
This paper adds a few comments and complements to the preceding one 1209; for instance, the enlargement map is bounded in $H^1$
Keywords: Enlargement of filtrations, Honest times
Nature: Original
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XII: 11, 78-97, LNM 649 (1978)
JEULIN, Thierry; YOR, Marc
Grossissement d'une filtration et semi-martingales~: Formules explicites (General theory of processes)
This contains very substantial improvements on 1209, namely, the explicit computation of the characteristics of the semimartingales involved
Comment: For additional results on enlargements, see the two Lecture Notes volumes 833 (T. Jeulin) and 1118. See also 1350
Keywords: Enlargement of filtrations, Honest times
Nature: Original
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XIII: 34, 400-406, LNM 721 (1979)
YOR, Marc
Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)
This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$
Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$
Nature: Original
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XIII: 50, 574-609, LNM 721 (1979)
JEULIN, Thierry
Grossissement d'une filtration et applications (General theory of processes, Markov processes)
This is a sequel to the papers 1209 and 1211, giving mostly applications of the theory of enlargements (turning a honest time $L$ into a stopping time) to Markov processes. The paper begins with a computation of conditional expectations relative to ${\cal F}_{L-}$, ${\cal F}_{L}$, ${\cal F}_{L+}$. This result is applied to coterminal times of a Markov process. Again a section is devoted to a general computation on two successive enlargements, which is shown to imply (with some work) Williams' well-known decomposition of Brownian paths
Keywords: Enlargement of filtrations, Williams decomposition
Nature: Original
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XIV: 20, 173-188, LNM 784 (1980)
MEYER, Paul-André
Les résultats de Jeulin sur le grossissement des tribus (General theory of processes, Stochastic calculus)
This is an introduction to beautiful results of Jeulin on enlargements, for which see Zeit. für W-Theorie, 52, 1980, and above all the Lecture Notes vol. 833, Semimartingales et grossissement d'une filtration
Keywords: Enlargement of filtrations, Semimartingales
Nature: Exposition
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XIV: 21, 189-199, LNM 784 (1980)
YOR, Marc
Application d'un lemme de Jeulin au grossissement de la filtration brownienne (General theory of processes, Brownian motion)
The problem considered here is the smallest enlargement of the Brownian filtration for which the process $\int_t^\infty B_s\mu(ds)$ is adapted, $\mu$ being a probability measure with a finite first moment
Comment: Note the misprint ${\cal G}$-martingale instead of ${\cal G}$-semimartingale in the statement of condition (H')
Keywords: Enlargement of filtrations
Nature: Original
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XV: 15, 210-226, LNM 850 (1981)
JEULIN, Thierry; YOR, Marc
Sur les distributions de certaines fonctionnelles du mouvement brownien (Brownian motion)
This paper gives new proofs and extensions of results due to Knight, concerning occupation times by the process $(S_t,B_t)$ up to time $T_a$, where $(B_t)$ is Brownian motion, $T_a$ the hitting time of $a$, and $(S_t)$ is $\sup_{s\le t} B_s$. The method uses enlargement of filtrations, and martingales similar to those of 1306. Theorem 3.7 is a decomposition of Brownian paths akin to Williams' decomposition
This paper is devoted to the a.s. absolute convergence of certain random integrals, a classical example of which is $\int_0^t ds/|B_s|^{\alpha}$ for Brownian motion starting from $0$. The author does not claim to prove deep results, but his technique of optional increasing reordering (réarrangement) of a process should be useful in other contexts too