VIII: 05, 25-26, LNM 381 (1974)
DELLACHERIE, Claude
Intégrales stochastiques par rapport aux processus de Wiener et de Poisson (
General theory of processes)
This paper shows that the previsible representation property of Brownian motion and the (compensated) Poisson processes is a consequence of the Wiener and Poisson measures being unique solutions of martingale problems
Comment: A gap in the proof is filled in
928 and
2002. This is a very important paper, opening the way to a series of investigations on the relations between previsible representation and extremality. See Jacod-Yor,
Z. für W-theorie, 38, 1977 and Yor
1221. For another approach to the restricted case considered here, see Ruiz de Chavez
1821. The previsible representation property of Brownian motion and compensated Poisson process was know by Itô; it is a consequence of the (stronger) chaotic representation property, established by Wiener in 1938. The converse was also known by Itô: among the martingales which are also Lévy processes, only Brownian motions and compensated Poisson processes have the previsible representation property
Keywords: Brownian motion,
Poisson processes,
Previsible representationNature: Original
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
XV: 43, 627-631, LNM 850 (1981)
WANG, Jia-Gang
Some remarks on processes with independent increments (
Independent increments)
This paper contains results on non-homogeneous processes with independent increments, without fixed discontinuities, which belong to the folklore of the subject but are hard to locate in the literature. The first one is that their natural filtration, merely augmented by all sets of measure $0$, is automatically right-continuous and quasi-left-continuous. The second one concerns those processes which are multivariate point processes, i.e., have only finitely many jumps in finite intervals and are constant between jumps. It is shown how to characterize the independent increments property into a property of the process of jumps conditioned by the process of jump times. Finally, a remark is done to the order that several results extend automatically to random measures with independent increments, for which see also
1544Keywords: Poisson processes,
Lévy measuresNature: Original Retrieve article from Numdam
XX: 02, 28-29, LNM 1204 (1986)
FAGNOLA, Franco;
LETTA, Giorgio
Sur la représentation intégrale des martingales du processus de Poisson (
Stochastic calculus,
Point processes)
Dellacherie gave in
805 a proof by stochastic calculus of the previsible representation property for the Wiener and Poisson processes. A gap in this proof is filled in
928 for Brownian motion and here for Poisson processes
Keywords: Stochastic integrals,
Previsible representation,
Poisson processesNature: Correction Retrieve article from Numdam
