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V: 33, 362-372, LNM 191 (1971)

**WEIL, Michel**

Conditionnement par rapport au passé strict (Markov processes)

Given a totally inaccessible terminal time $T$, it is shown how to compute conditional expectations of the future with respect to the strict past $\sigma$-field ${\cal F}_{T-}$. The formula involves the Lévy system of the process

Comment: B. Maisonneuve pointed out once that the paper, though essentially correct, has a small mistake somewhere. See Dellacherie-Meyer,*Probabilité et Potentiels,* Chap. XX **46**--48

Keywords: Terminal times, Lévy systems

Nature: Original

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VII: 01, 1-24, LNM 321 (1973)

**BENVENISTE, Albert**

Application de deux théorèmes de G.~Mokobodzki à l'étude du noyau de Lévy d'un processus de Hunt sans hypothèse (L) (Markov processes)

The object of the theory of Lévy systems is to compute the previsible compensator of sums $\sum_{s\le t} f(X_{s-},X_s)$ extended to the jump times of a Markov process~$X$, i.e., the times $s$ at which $X_s\not=X_{s-}$. The theory was created by Lévy in the case of a process with independent increments, and the classical results for Markov processes are due to Ikeda-Watanabe,*J. Math. Kyoto Univ.*, **2**, 1962 and Watanabe, *Japan J. Math.*, **34**, 1964. An exposition of their results can be found in the Seminar, 106. The standard assumptions were: 1) $X$ is a Hunt process, implying that jumps occur at totally inaccessible stopping times and the compensator is continuous, 2) Hypothesis (L) (absolute continuity of the resolvent) is satisfied. Here using two results of Mokobodzki: 1) every excessive function dominated in the strong sense in a potential. 2) The existence of medial limits (this volume, 719), Hypothesis (L) is shown to be unnecessary

Comment: Mokobodzki's second result depends on additional axioms in set theory, the continuum hypothesis or Martin's axiom. See also Benveniste-Jacod,*Invent. Math.* **21**, 1973, which no longer uses medial limits

Keywords: Lévy systems, Additive functionals

Nature: Original

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VII: 02, 25-32, LNM 321 (1973)

**MEYER, Paul-André**

Une mise au point sur les systèmes de Lévy. Remarques sur l'exposé de A. Benveniste (Markov processes)

This is an addition to the preceding paper 701, extending the theory to right processes by means of a Ray compactification

Comment: All this material has become classical. See for instance Dellacherie-Meyer,*Probabilités et Potentiel,* vol. D, chapter XV, 31--35

Keywords: Lévy systems, Ray compactification

Nature: Original

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X: 22, 481-500, LNM 511 (1976)

**YOR, Marc**

Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles (Martingale theory, Stochastic calculus)

This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula ${\cal E}(X)\,{\cal E}(Y)={\cal E}(X+Y+[X,Y])$ where ${\cal E}$ denotes the usual exponential of semimartingales. Also, the s.d.e. $Z_t=1+\int_0^t Z_sdX_s$ is solved, where $X$ is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of ``exponentials'' is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita-Watanabe in a Markovian set-up

Keywords: Optional stochastic integrals, Stochastic exponentials, Lévy systems

Nature: Original

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XI: 37, 529-538, LNM 581 (1977)

**MAISONNEUVE, Bernard**

Changement de temps d'un processus markovien additif (Markov processes)

A Markov additive process $(X_t,S_t)$ (Cinlar,*Z. für W-theorie,* **24**, 1972) is a generalisation of a pair $(X,S)$ where $X$ is a Markov process with arbitrary state space, and $S$ is an additive functional of $X$: in the general situation $S$ is positive real valued, $X$ is a Markov process in itself, and the pair $(X,S)$ is a Markov processes, while $S$ is an additive functional *of the pair.* For instance, subordinators are Markov additive processes with trivial $X$. A simpler proof of a basic formula of Cinlar is given, and it is shown also that a Markov additive process gives rise to a regenerative system in a slightly extended sense

Comment: See also 1513

Keywords: Markov additive processes, Additive functionals, Regenerative sets, Lévy systems

Nature: Original

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Conditionnement par rapport au passé strict (Markov processes)

Given a totally inaccessible terminal time $T$, it is shown how to compute conditional expectations of the future with respect to the strict past $\sigma$-field ${\cal F}_{T-}$. The formula involves the Lévy system of the process

Comment: B. Maisonneuve pointed out once that the paper, though essentially correct, has a small mistake somewhere. See Dellacherie-Meyer,

Keywords: Terminal times, Lévy systems

Nature: Original

Retrieve article from Numdam

VII: 01, 1-24, LNM 321 (1973)

Application de deux théorèmes de G.~Mokobodzki à l'étude du noyau de Lévy d'un processus de Hunt sans hypothèse (L) (Markov processes)

The object of the theory of Lévy systems is to compute the previsible compensator of sums $\sum_{s\le t} f(X_{s-},X_s)$ extended to the jump times of a Markov process~$X$, i.e., the times $s$ at which $X_s\not=X_{s-}$. The theory was created by Lévy in the case of a process with independent increments, and the classical results for Markov processes are due to Ikeda-Watanabe,

Comment: Mokobodzki's second result depends on additional axioms in set theory, the continuum hypothesis or Martin's axiom. See also Benveniste-Jacod,

Keywords: Lévy systems, Additive functionals

Nature: Original

Retrieve article from Numdam

VII: 02, 25-32, LNM 321 (1973)

Une mise au point sur les systèmes de Lévy. Remarques sur l'exposé de A. Benveniste (Markov processes)

This is an addition to the preceding paper 701, extending the theory to right processes by means of a Ray compactification

Comment: All this material has become classical. See for instance Dellacherie-Meyer,

Keywords: Lévy systems, Ray compactification

Nature: Original

Retrieve article from Numdam

X: 22, 481-500, LNM 511 (1976)

Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles (Martingale theory, Stochastic calculus)

This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula ${\cal E}(X)\,{\cal E}(Y)={\cal E}(X+Y+[X,Y])$ where ${\cal E}$ denotes the usual exponential of semimartingales. Also, the s.d.e. $Z_t=1+\int_0^t Z_sdX_s$ is solved, where $X$ is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of ``exponentials'' is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita-Watanabe in a Markovian set-up

Keywords: Optional stochastic integrals, Stochastic exponentials, Lévy systems

Nature: Original

Retrieve article from Numdam

XI: 37, 529-538, LNM 581 (1977)

Changement de temps d'un processus markovien additif (Markov processes)

A Markov additive process $(X_t,S_t)$ (Cinlar,

Comment: See also 1513

Keywords: Markov additive processes, Additive functionals, Regenerative sets, Lévy systems

Nature: Original

Retrieve article from Numdam