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22 matches found
X: 03, 24-39, LNM 511 (1976)
JACOD, Jean; MÉMIN, Jean
Un théorème de représentation des martingales pour les ensembles régénératifs (Martingale theory, Markov processes, Stochastic calculus)
The natural filtration of a regenerative set $M$ is that of the corresponding age process''. There is a natural optional random measure $\mu$ carried by the right endppoints of intervals contiguous to $M$, each endpoint carrying a mass equal to the length of its interval. Let $\nu$ be the previsible compensator of $\mu$. It is shown that, if $M$ has an empty interior the martingale measure $\mu-\nu$ has the previsible representation property in the natural filtration
Comment: Martingales in the filtration of a random set (not necessarily regenerative) have been studied by Azéma in 1932. In the case of the set of zeros of Brownian motion, the martingale considered here is the second Azéma's martingale'' (not the well known one which has the chaotic representation property)
Keywords: Regenerative sets, Renewal theory, Stochastic integrals, Previsible representation
Nature: Original
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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 1530
Keywords: Stochastic integrals, Optional stochastic integrals, Random measures, Semimartingales
Nature: Original
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XII: 03, 22-34, LNM 649 (1978)
JACOD, Jean
Projection prévisible et décomposition multiplicative d'une semi-martingale positive (General theory of processes)
The problem discussed is the decomposition of a positive ($\ge0$) special semimartingale $X$ (the most interesting cases being super- and submartingales) into a product of a positive local martingale and a positive previsible process of finite variation. The problem is solved here in the greatest possible generality, on a maximal non-vanishing domain for $X$---this is a previsible stochastic interval $[0,S)$ which at $S$ may be open or closed
Comment: This papers improves on 1021 and 1023
Keywords: Semimartingales, Multiplicative decomposition
Nature: Original
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XIV: 19, 161-172, LNM 784 (1980)
JACOD, Jean
Intégrales stochastiques par rapport à une semi-martingale vectorielle et changements de filtration (Stochastic calculus, General theory of processes)
Given a square integrable vector martingale $M$ and a previsible vector process $H$, the conditions implying the existence of the (scalar valued) stochastic integral $H.M$ are less restrictive than the existence of the componentwise'' stochastic integral, unless the components of $M$ are orthogonal (this result was due to Galtchouk, 1975). The theory of vector stochastic integrals, though parallel to the scalar theory, requires a careful theory given in this paper
Comment: Another approach, yielding an equivalent definition, is followed by L. Schwartz in his article 1530 on formal semimartingales
Keywords: Semimartingales, Stochastic integrals
Nature: Original
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XIV: 27, 227-248, LNM 784 (1980)
JACOD, Jean; MÉMIN, Jean
Sur la convergence des semimartingales vers un processus à accroissements indépendants (General theory of processes, Stochastic calculus, Martingale theory)
A method of Kabanov, Liptzer and Shiryaev is adapted to study the convergence of a sequence of semimartingales to a process with independent increments (to be completed)
Keywords: Convergence in law, Tightness
Nature: Original
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XIV: 43, 410-417, LNM 784 (1980)
JACOD, Jean; MAISONNEUVE, Bernard
Remarque sur les fonctionnelles additives non adaptées des processus de Markov (Markov processes)
It occurs sometimes that a Markov process $(X_t)$ satisfies in a filtration ${\cal H}_t$ a Markov property of the form $E[f\circ \theta_t \,|\,{\cal H}_t]= E_{X_t}[f]$, where $f$ is not restricted to be ${\cal H}_t$-measurable. For instance, situations in renewal theory where one is given a Markov pair $(X_t,Y_t)$, and ${\cal H}_t$ describes the path of $X$ up to time $t$, and the whole path of $Y$. In such cases, the authors show that additive functionals which are previsible in the larger filtration are in fact previsible in the filtration of $X$ alone
Nature: Original
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XV: 36, 529-546, LNM 850 (1981)
JACOD, Jean; MÉMIN, Jean
Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité (Measure theory)
For simplicity we consider only real valued r.v.'s, but it is essential that the paper considers general Polish spaces instead of $R$. Let us define a fuzzy r.v. $X$ on $(\Omega, {\cal F},P)$ as a probability measure on $\Omega\timesR$ whose projection on $\Omega$ is $P$. In particular, a standard r.v. $X$ defines such a measure as the image of $P$ under the map $\omega\mapsto (\omega,X(\omega))$. The space of fuzzy r.v.'s is provided with a weak topology, associated with the bounded functions $f(\omega,x)$ which are continuous in $x$ for every $\omega$, or equivalently with the functions $I_A(\omega)\,f(x)$ with $f$ bounded continuous. The main topic of this paper is the study of this topology
Comment: From this description, it is clear that this paper extends to general Polish spaces the topology of Baxter-Chacon (forgetting about the filtration), for which see 1228
Keywords: Fuzzy random variables, Convergence in law
Nature: Original
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XV: 37, 547-560, LNM 850 (1981)
JACOD, Jean
Convergence en loi de semimartingales et variation quadratique (General theory of processes, Stochastic calculus)
The convergence in law of cadlag processes to a cadlag process being understood in the sense of Skorohod, the problem is to find sufficient conditions under which, given semimartingales $X^n$ and $X$ such that $X^n\rightarrow X$ in law, one may deduce that $[X^n,X^n]$ converges in law to $[X,X]$. This is achieved assuming a uniform bound on the expectations of the supremum of the jumps. A version of the theorem applied to processes which are not semimartingales, but are equal to semimartingales on large sets
Keywords: Semimartingales, Skorohod topology, Convergence in law
Nature: Original
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XVI: 39, 442-446, LNM 920 (1982)
JACOD, Jean
Équations différentielles linéaires : la méthode de variation des constantes
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XVI: 40, 447-450, LNM 920 (1982)
JACOD, Jean; PROTTER, Philip
Quelques remarques sur un nouveau type d'équations différentielles stochastiques
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XVII: 15, 132-157, LNM 986 (1983)
BICHTELER, Klaus; JACOD, Jean
Calcul de Malliavin pour les diffusions avec sauts : Existence d'une densité dans le cas unidimensionnel
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XVII: 44, 509-511, LNM 986 (1983)
JACOD, Jean; MÉMIN, Jean
Rectification à Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité
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XVIII: 08, 91-118, LNM 1059 (1984)
JACOD, Jean
Une généralisation des semimartingales : les processus admettant un processus à accroissements indépendants tangent
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XXI: 28, 479-514, LNM 1247 (1987)
Processus admettant un processus à accroissements indépendants tangent : Cas général
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XXIII: 37, 448-474, LNM 1372 (1989)
JACOD, Jean
Une application de la topologie d'Émery : le processus information d'un modèle statistique filtré
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XXIV: 18, 282-299, LNM 1426 (1990)
COQUET, François; JACOD, Jean
Convergence des surmartingales --- Application aux vraisemblances partielles
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XXV: 14, 138-139, LNM 1485 (1991)
JACOD, Jean; PROTTER, Philip
Une remarque sur les équations différentielles stochastiques à solutions markoviennes
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XXV: 15, 140-161, LNM 1485 (1991)
JACOD, Jean
Régularité d'ordre quelconque pour un modèle statistique filtré
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XXVIII: 02, 21-35, LNM 1583 (1994)