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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 *

Comment: See also 1329, 1350

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

Comment: See also 1516

Keywords: Explicit laws, Occupation times, Enlargement of filtrations, Williams decomposition

Nature: Original

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XVI: 22, 248-256, LNM 920 (1982)

**JEULIN, Thierry**

Sur la convergence absolue de certaines intégrales (General theory of processes)

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

Comment: This paper greatly simplifies a proof in the author's*Semimartingales et Grossissement de Filtrations,* LNM **833**, p.44

Keywords: Enlargement of filtrations

Nature: Original

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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

Retrieve article from Numdam

XII: 09, 61-69, LNM 649 (1978)

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

Comment: This result was independently discovered by Barlow,

Keywords: Enlargement of filtrations, Honest times

Nature: Original

Retrieve article from Numdam

XII: 10, 70-77, LNM 649 (1978)

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

Retrieve article from Numdam

XII: 11, 78-97, LNM 649 (1978)

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

Keywords: Enlargement of filtrations, Honest times

Nature: Original

Retrieve article from Numdam

XIII: 34, 400-406, LNM 721 (1979)

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

Retrieve article from Numdam

XIII: 50, 574-609, LNM 721 (1979)

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

Retrieve article from Numdam

XIV: 20, 173-188, LNM 784 (1980)

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

Comment: See also 1329, 1350

Keywords: Enlargement of filtrations, Semimartingales

Nature: Exposition

Retrieve article from Numdam

XIV: 21, 189-199, LNM 784 (1980)

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

Retrieve article from Numdam

XV: 15, 210-226, LNM 850 (1981)

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

Comment: See also 1516

Keywords: Explicit laws, Occupation times, Enlargement of filtrations, Williams decomposition

Nature: Original

Retrieve article from Numdam

XVI: 22, 248-256, LNM 920 (1982)

Sur la convergence absolue de certaines intégrales (General theory of processes)

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

Comment: This paper greatly simplifies a proof in the author's

Keywords: Enlargement of filtrations

Nature: Original

Retrieve article from Numdam