Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 ``Probabilités et Modèles Aléatoires''

### On processes with conditional independent increments and stable convergence in law

Auteur(s):

Code(s) de Classification MSC:

• 60F17 Functional limit theorems; invariance principles
• 60H99 None of the above but in this section

Résumé: In this paper we study the semiamrtingales \$X\$ which are defined on an extension of a basic filtered probability space \$\Ba=(\Om,\Fa,\fit_{t\geq0},P)\$ and which, conditionally on \$\Fa\$, have independent increments. We first give a general characterization for such processes. Then we prove that if all martingales of the basis \$\Ba\$ can be written as a sum of stochastic integrals w.r.t. the continuous martingale part and the compensated jump measure of \$Y\$, then a process \$X\$ has \$\Fa\$-conditional independent increments if and only if the characteristics of the pair \$(X,Y)\$, on the extended space, are indeed predictable w.r.t. the filtration \$\fit\$. Finally we prove a functional convergence result toward a process \$X\$ of this kind.

Mots Clés: Stable convergence ; Lévy processes

Date: 2001-05-21

Prépublication numéro: PMA-662