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VIII: 18, 316-328, LNM 381 (1974)

**PRIOURET, Pierre**

Construction de processus de Markov sur ${\bf R}^n$ (Markov processes, Diffusion theory)

The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic

Comment: To be completed

Keywords: Processes with jumps

Nature: Original

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XI: 21, 356-361, LNM 581 (1977)

**CHOU, Ching Sung**

Le processus des sauts d'une martingale locale (Martingale theory)

Simple necessary and sufficient conditions are given on an optional process $\sigma_t$ (different from $0$ only at countably many stopping times) so that it is the process of jumps $ėlta M_t$ of some local martingale $M$

Comment: The same result is proved independently by D. Lépingle in this volume, see 1129. For an application see 1308, and for another approach 1335

Keywords: Local martingales, Jumps

Nature: Original

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XI: 29, 418-434, LNM 581 (1977)

**LÉPINGLE, Dominique**

Sur la représentation des sauts des martingales (Martingale theory)

The problem discussed in this paper consists in decomposing into two parts a local martingale, so that one part has its jumps contained in a given thin optional set $D$ and the other one is continuous on $D$. The main theorem of 1121 is proved independently as an important technical tool

Comment: See also 1335

Keywords: Local martingales, Jumps, Optional stochastic integrals

Nature: Original

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XIII: 37, 441-442, LNM 721 (1979)

**CHOU, Ching Sung**

Démonstration simple d'un résultat sur le temps local (Stochastic calculus)

It follows from Ito's formula that the positive parts of those jumps of a semimartingale $X$ that originate below $0$ are summable. A direct proof is given of this fact

Comment: Though the idea is essentially correct, an embarrassing mistake is corrected as 1429

Keywords: Local times, Semimartingales, Jumps

Nature: Original

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XIV: 29, 254-254, LNM 784 (1980)

**YOEURP, Chantha**

Rectificatif à l'exposé de C.S. Chou (Stochastic calculus)

A mistake in the proof of 1337 is corrected, the result remaining true without additional assumptions

Keywords: Local times, Semimartingales, Jumps

Nature: Correction

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XIV: 30, 255-255, LNM 784 (1980)

**REBOLLEDO, Rolando**

Corrections à ``Décomposition des martingales locales et raréfaction des sauts'' (General theory of processes, Martingale theory)

Concerns 1311. For the definitive version, see*Mém. Soc. Math. France,* **62**, 1979

Keywords: Central limit theorem, Skorohod topology, Local martingales, Jumps

Nature: Correction

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XV: 41, 604-617, LNM 850 (1981)

**LÉPINGLE, Dominique**; **MEYER, Paul-André**; **YOR, Marc**

Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)

This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case

Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps

Nature: Original

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XXV: 18, 196-219, LNM 1485 (1991)

**PICARD, Jean**

Calcul stochastique avec sauts sur une variété (Stochastic differential geometry)

It is known from Meyer 1505 that intrinsic Ito integrals have a meaning for continuous semimartingales in a manifold $M$, provided $M$ is endowed with a connection. This is extended here to càdlàg semimartingales. The manifold must be endowed with a richer structure, a ``connector'', mapping $M\times M$ to the tangent bundle, that allows to interpret a jump $(X_{t-},X_t)$ as a tangent vector to $M$ at $X{t-}$; the differential of the connector at the diagonal reduces to a classical torsion-free connection. Introducing torsions leads to a more general ``transporter'', describing how parallel transports should behave at jump times, and reducing to a classical connection for infinitesimal jumps. Discrete-time approximations are established.

Keywords: Semimartingales in manifolds, Martingales in manifolds, Jumps

Nature: Original

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XXVI: 11, 127-145, LNM 1526 (1992)

**ESTRADE, Anne**; **PONTIER, Monique**

Relèvement horizontal d'une semimartingale càdlàg (Stochastic differential geometry, Stochastic calculus)

For filtering purposes, the lifting of a manifold-valued semimartingale $X$ to the tangent space at $X_0$ is extended here to the case when $X$ has jumps. The value of $L_t$ involves the inverse of the exponential at $X_{t-}$ applied to $X_t$, and a parallel transport from $X_0$ to $X_{t-}$

Comment: The same method is described in a more general setting by Kurtz-Pardoux-Protter*Ann I.H.P.* (1995). In turn, this is a particular instance of a very general scheme due to Cohen (*Stochastics Stoch. Rep.* (1996)

Keywords: Stochastic parallel transport, Stochastic differential equations, Jumps

Nature: Original

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XXIX: 16, 166-180, LNM 1613 (1995)

**APPLEBAUM, David**

A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (Stochastic differential geometry, Markov processes)

This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.

Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (*Stochastics Stochastics Rep.* **56**, 1996). The same question is addressed by Cohen in the next article 2917

Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators

Nature: Original

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XXIX: 17, 181-193, LNM 1613 (1995)

**COHEN, Serge**

Some Markov properties of stochastic differential equations with jumps (Stochastic differential geometry, Markov processes)

The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see 1505 and 1655) was extended by Cohen to càdlàg semimartingales (*Stochastics Stochastics Rep.* **56**, 1996). Here this language is used to study the Markov property of solutions to SDE's with jumps. In particular,two definitions of a Lévy process in a Riemannian manifold are compared: One as the solution to a SDE driven by some Euclidean Lévy process, the other by subordinating some Riemannian Brownian motion. It is shown that in general the former is not of the second kind

Comment: The first definition is independently introduced by David Applebaum 2916

Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators

Nature: Original

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XLIV: 20, 429-465, LNM 2046 (2012)

**LÉONARD, Christian**

Girsanov theory under a finite entropy condition (Theory of processes)

Keywords: Stochastic processes, Relative entropy, Girsanov's theory, Diffusion processes, Processes with jumps

Nature: Original

Construction de processus de Markov sur ${\bf R}^n$ (Markov processes, Diffusion theory)

The problem is to construct a Markov process which satisfies a martingale problem relative to a generator involving a diffusion part and a jump part. The method used is analytic

Comment: To be completed

Keywords: Processes with jumps

Nature: Original

Retrieve article from Numdam

XI: 21, 356-361, LNM 581 (1977)

Le processus des sauts d'une martingale locale (Martingale theory)

Simple necessary and sufficient conditions are given on an optional process $\sigma_t$ (different from $0$ only at countably many stopping times) so that it is the process of jumps $ėlta M_t$ of some local martingale $M$

Comment: The same result is proved independently by D. Lépingle in this volume, see 1129. For an application see 1308, and for another approach 1335

Keywords: Local martingales, Jumps

Nature: Original

Retrieve article from Numdam

XI: 29, 418-434, LNM 581 (1977)

Sur la représentation des sauts des martingales (Martingale theory)

The problem discussed in this paper consists in decomposing into two parts a local martingale, so that one part has its jumps contained in a given thin optional set $D$ and the other one is continuous on $D$. The main theorem of 1121 is proved independently as an important technical tool

Comment: See also 1335

Keywords: Local martingales, Jumps, Optional stochastic integrals

Nature: Original

Retrieve article from Numdam

XIII: 37, 441-442, LNM 721 (1979)

Démonstration simple d'un résultat sur le temps local (Stochastic calculus)

It follows from Ito's formula that the positive parts of those jumps of a semimartingale $X$ that originate below $0$ are summable. A direct proof is given of this fact

Comment: Though the idea is essentially correct, an embarrassing mistake is corrected as 1429

Keywords: Local times, Semimartingales, Jumps

Nature: Original

Retrieve article from Numdam

XIV: 29, 254-254, LNM 784 (1980)

Rectificatif à l'exposé de C.S. Chou (Stochastic calculus)

A mistake in the proof of 1337 is corrected, the result remaining true without additional assumptions

Keywords: Local times, Semimartingales, Jumps

Nature: Correction

Retrieve article from Numdam

XIV: 30, 255-255, LNM 784 (1980)

Corrections à ``Décomposition des martingales locales et raréfaction des sauts'' (General theory of processes, Martingale theory)

Concerns 1311. For the definitive version, see

Keywords: Central limit theorem, Skorohod topology, Local martingales, Jumps

Nature: Correction

Retrieve article from Numdam

XV: 41, 604-617, LNM 850 (1981)

Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)

This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case

Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps

Nature: Original

Retrieve article from Numdam

XXV: 18, 196-219, LNM 1485 (1991)

Calcul stochastique avec sauts sur une variété (Stochastic differential geometry)

It is known from Meyer 1505 that intrinsic Ito integrals have a meaning for continuous semimartingales in a manifold $M$, provided $M$ is endowed with a connection. This is extended here to càdlàg semimartingales. The manifold must be endowed with a richer structure, a ``connector'', mapping $M\times M$ to the tangent bundle, that allows to interpret a jump $(X_{t-},X_t)$ as a tangent vector to $M$ at $X{t-}$; the differential of the connector at the diagonal reduces to a classical torsion-free connection. Introducing torsions leads to a more general ``transporter'', describing how parallel transports should behave at jump times, and reducing to a classical connection for infinitesimal jumps. Discrete-time approximations are established.

Keywords: Semimartingales in manifolds, Martingales in manifolds, Jumps

Nature: Original

Retrieve article from Numdam

XXVI: 11, 127-145, LNM 1526 (1992)

Relèvement horizontal d'une semimartingale càdlàg (Stochastic differential geometry, Stochastic calculus)

For filtering purposes, the lifting of a manifold-valued semimartingale $X$ to the tangent space at $X_0$ is extended here to the case when $X$ has jumps. The value of $L_t$ involves the inverse of the exponential at $X_{t-}$ applied to $X_t$, and a parallel transport from $X_0$ to $X_{t-}$

Comment: The same method is described in a more general setting by Kurtz-Pardoux-Protter

Keywords: Stochastic parallel transport, Stochastic differential equations, Jumps

Nature: Original

Retrieve article from Numdam

XXIX: 16, 166-180, LNM 1613 (1995)

A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (Stochastic differential geometry, Markov processes)

This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.

Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (

Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators

Nature: Original

Retrieve article from Numdam

XXIX: 17, 181-193, LNM 1613 (1995)

Some Markov properties of stochastic differential equations with jumps (Stochastic differential geometry, Markov processes)

The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see 1505 and 1655) was extended by Cohen to càdlàg semimartingales (

Comment: The first definition is independently introduced by David Applebaum 2916

Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators

Nature: Original

Retrieve article from Numdam

XLIV: 20, 429-465, LNM 2046 (2012)

Girsanov theory under a finite entropy condition (Theory of processes)

Keywords: Stochastic processes, Relative entropy, Girsanov's theory, Diffusion processes, Processes with jumps

Nature: Original