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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: 29, 332-359, LNM 721 (1979)

**JEULIN, Thierry**; **YOR, Marc**

Inégalité de Hardy, semimartingales, et faux-amis (Martingale theory, General theory of processes)

The main purpose of this paper is to warn against ``obvious'' statements which are in fact false. Let $({\cal G}_t)$ be an enlargement of $({\cal F}_t)$. Assume that ${\cal F}$ has the previsible representation property with respect to a martingale $X$ which is a ${\cal G}$-semimartingale. Then it does not follow that every ${\cal F}$-martingale $Y$ is a ${\cal G}$-semimartingale. Also, even if $Y$ is a ${\cal G}$-semimartingale, its ${\cal G}$-compensator may have bad absolute continuity properties. The counterexample to the first statement involves a detailed study of the initial enlargement of the filtration of Brownian motion $(B_t)_{t\le 1}$ by the random variable $B_1$, which transforms it into the Brownian bridge, a semimartingale. Then the stochastic integrals with respect to $B$ which are ${\cal G}$-semimartingales are completely described, and this is the place where the classical Hardy inequality appears

Keywords: Hardy's inequality, Previsible representation

Nature: Original

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XIII: 30, 360-370, LNM 721 (1979)

**JEULIN, Thierry**; **YOR, Marc**

Sur l'expression de la dualité entre $H^1$ et $BMO$ (Martingale theory)

The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$

Comment: On the same topic see 1518

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

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XIII: 45, 521-532, LNM 721 (1979)

**JEULIN, Thierry**

Un théorème de J.W. Pitman (Brownian motion, Diffusion theory)

This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$

Keywords: Bessel processes

Nature: New proof of known results

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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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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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XXIV: 15, 227-265, LNM 1426 (1990)

**JEULIN, Thierry**; **YOR, Marc**

Filtration des ponts browniens et équations différentielles stochastiques linéaires

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XXVI: 24, 322-347, LNM 1526 (1992)

**JEULIN, Thierry**; **YOR, Marc**

Une décomposition non-canonique du drap brownien (Brownian sheet, Gaussian processes)

In 2415, the authors have introduced a transform of Brownian motion. Here, a similar transform is defined on the Brownian sheet; this transform is shown to be strongly mixing

Comment: This work was motivated by Föllmer's article on Martin boundaries on Wiener space (in*Diffusion processes and related problems in analysis*, vol.~I, Birkhäuser 1990)

Keywords: Brownian motion, Several parameter processes

Nature: Original

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XXVII: 08, 53-77, LNM 1557 (1993)

**JEULIN, Thierry**; **YOR, Marc**

Moyennes mobiles et semimartingales

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XXVII: 15, 133-158, LNM 1557 (1993)

**AZÉMA, Jacques**; **JEULIN, Thierry**; **KNIGHT, Frank B.**; **YOR, Marc**

Le théorème d'arrêt en une fin d'ensemble prévisible

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XXX: 20, 312-343, LNM 1626 (1996)

**AZÉMA, Jacques**; **JEULIN, Thierry**; **KNIGHT, Frank B.**; **MOKOBODZKI, Gabriel**; **YOR, Marc**

Sur les processus croissants de type injectif

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XXXII: 22, 316-327, LNM 1686 (1998)

**AZÉMA, Jacques**; **JEULIN, Thierry**; **KNIGHT, Frank B.**; **YOR, Marc**

Quelques calculs de compensateurs impliquant l'injectivité de certains processus croissants

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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: 29, 332-359, LNM 721 (1979)

Inégalité de Hardy, semimartingales, et faux-amis (Martingale theory, General theory of processes)

The main purpose of this paper is to warn against ``obvious'' statements which are in fact false. Let $({\cal G}_t)$ be an enlargement of $({\cal F}_t)$. Assume that ${\cal F}$ has the previsible representation property with respect to a martingale $X$ which is a ${\cal G}$-semimartingale. Then it does not follow that every ${\cal F}$-martingale $Y$ is a ${\cal G}$-semimartingale. Also, even if $Y$ is a ${\cal G}$-semimartingale, its ${\cal G}$-compensator may have bad absolute continuity properties. The counterexample to the first statement involves a detailed study of the initial enlargement of the filtration of Brownian motion $(B_t)_{t\le 1}$ by the random variable $B_1$, which transforms it into the Brownian bridge, a semimartingale. Then the stochastic integrals with respect to $B$ which are ${\cal G}$-semimartingales are completely described, and this is the place where the classical Hardy inequality appears

Keywords: Hardy's inequality, Previsible representation

Nature: Original

Retrieve article from Numdam

XIII: 30, 360-370, LNM 721 (1979)

Sur l'expression de la dualité entre $H^1$ et $BMO$ (Martingale theory)

The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$

Comment: On the same topic see 1518

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

Retrieve article from Numdam

XIII: 45, 521-532, LNM 721 (1979)

Un théorème de J.W. Pitman (Brownian motion, Diffusion theory)

This paper contains an appendix by M. Yor. Let $(B_t)$ and $(Z_t)$ be a Brownian motion and a Bes$_3$ process both starting at $0$. Put $S_t=\sup_{s\le t} B_t$ and $J_t=\inf_{s\ge t}Z_t$. Then Pitman's theorem asserts that, in law, $2S-B=Z$ and $2J-Z=B$ (both statements being in fact equivalent). A complete proof of the theorem is given, using techniques from the general theory of processes. The appendix shows that, granted that $2S-B$ is Markov, it is easy to see that it is a Bes$_3$

Keywords: Bessel processes

Nature: New proof of known results

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

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

XXIV: 15, 227-265, LNM 1426 (1990)

Filtration des ponts browniens et équations différentielles stochastiques linéaires

Retrieve article from Numdam

XXVI: 24, 322-347, LNM 1526 (1992)

Une décomposition non-canonique du drap brownien (Brownian sheet, Gaussian processes)

In 2415, the authors have introduced a transform of Brownian motion. Here, a similar transform is defined on the Brownian sheet; this transform is shown to be strongly mixing

Comment: This work was motivated by Föllmer's article on Martin boundaries on Wiener space (in

Keywords: Brownian motion, Several parameter processes

Nature: Original

Retrieve article from Numdam

XXVII: 08, 53-77, LNM 1557 (1993)

Moyennes mobiles et semimartingales

Retrieve article from Numdam

XXVII: 15, 133-158, LNM 1557 (1993)

Le théorème d'arrêt en une fin d'ensemble prévisible

Retrieve article from Numdam

XXX: 20, 312-343, LNM 1626 (1996)

Sur les processus croissants de type injectif

Retrieve article from Numdam

XXXII: 22, 316-327, LNM 1686 (1998)

Quelques calculs de compensateurs impliquant l'injectivité de certains processus croissants

Retrieve article from Numdam