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 jumpsNature: Original
Retrieve article from Numdam
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
1335Keywords: Local martingales,
JumpsNature: Original Retrieve article from Numdam
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
1335Keywords: Local martingales,
Jumps,
Optional stochastic integralsNature: Original Retrieve article from Numdam
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
1429Keywords: Local times,
Semimartingales,
JumpsNature: Original Retrieve article from Numdam
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,
JumpsNature: Correction Retrieve article from Numdam
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,
JumpsNature: Correction Retrieve article from Numdam
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,
JumpsNature: Original Retrieve article from Numdam
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,
JumpsNature: Original Retrieve article from Numdam
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,
JumpsNature: Original Retrieve article from Numdam
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
2917Keywords: Semimartingales with jumps,
Lévy processes,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
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
2916Keywords: Semimartingales with jumps,
Lévy processes,
Subordination,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
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 jumpsNature: Original
