III: 15, 190-229, LNM 88 (1969)
MORANDO, Philippe
Mesures aléatoires (
Independent increments)
This paper consists of two talks, on the construction and structure of measures with independent values on an abstract measurable space, inspired by papers of Prekopa (
Acta Math. Acad. Sci. Hung., 7, 1956 and
8, 1957) and Kingman (
Pacific J. Math., 21, 1967)
Comment: If the measurable space is not ``too'' abstract, it can be imbedded into the line, and the standard theory of Lévy processes (non-homogeneous) can be used. This simple remark reduces the interest of the general treatment: see Dellacherie-Meyer,
Probabilités et potentiel, Chapter XIII, end of \S4
Keywords: Random measures,
Independent incrementsNature: Exposition,
Original additions
Retrieve article from Numdam
IX: 30, 496-514, LNM 465 (1975)
SHARPE, Michael J.
Homogeneous extensions of random measures (
Markov processes)
Homogeneous random measures are the appropriate definition of additive functionals which may explode. The problem discussed here is the extension of such a measure given up to a terminal time into a measure defined up to the lifetime
Comment: The subject is taken over in a systematic way in Sharpe,
General Theory of Markov processes, Academic Press 1988
Keywords: Homogeneous random measures,
Terminal times,
SubprocessesNature: Original Retrieve article from Numdam
XI: 26, 390-410, LNM 581 (1977)
JACOD, Jean
Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales (
Martingale theory)
This paper develops the theory of stochastic integration (previsible and optional) with respect to local martingales starting from the particular case of continuous local martingales, and from the explicit description of the jumps of a local martingale (
1121,
1129). Then the theory of stable subspaces of $H^1$ (instead of the usual $H^2$) is developed, as well as the stochastic integral with respect to a random measure. A characterization is given of the jump process of a semimartingale. Then previsible stochastic integrals for semimartingales are given a maximal extension, and optional integrals for semimartingales (differing as usual from those for martingales) are defined
Comment: On the maximal extension of the stochastic integral $H{\cdot}X$ with $H$ previsible, see also Jacod,
Calcul stochastique et problèmes de martingales, Springer 1979. Other, equivalent, definitions are given in
1415,
1417,
1424 and
1530Keywords: Stochastic integrals,
Optional stochastic integrals,
Random measures,
SemimartingalesNature: Original Retrieve article from Numdam
XIII: 15, 199-203, LNM 721 (1979)
MEYER, Paul-André
Une remarque sur le calcul stochastique dépendant d'un paramètre (
General theory of processes)
Call a ``process'' a measurable function $X(u,t,\omega)$ where $t$ and $\omega$ are as usual and $u$ is a parameter ranging over some nice measurable space ${\cal U}$. Say that $X$ is evanescent if $X(.\,,\,.\,,\omega)\equiv0$ for a.a. $\omega$. The problem is to define previsible processes, and previsible projections defined up to evanescent sets. This is achieved following Jacod,
Zeit. für W-Theorie, 31, 1975. The main feature is the corresponding use of random measures, previsible random measures, and previsible dual projections
Keywords: Processes depending on a parameter,
Previsible processes,
Previsible projections,
Random measuresNature: Exposition,
Original additions Retrieve article from Numdam
XIII: 16, 204-215, LNM 721 (1979)
DOLÉANS-DADE, Catherine;
MEYER, Paul-André
Un petit théorème de projection pour processus à deux indices (
Several parameter processes)
This paper proves a previsible projection theorem in the case of two-parameter processes, with a two-parameter filtration satisfying the standard commutation property of Cairoli-Walsh. The idea is to apply successively the projection operation of
1315 to the two coordinates
Keywords: Previsible processes (several parameters),
Previsible projections,
Random measuresNature: Original Retrieve article from Numdam
