On processes with conditional independent increments and stable convergence in law

In this paper we study the semimartingales X which are defined on an extension of a basic filtered probability space B = (Omega,F,(F(t))(tgreater than or equal to0),P) and which, conditionally on F, have independent increments. We first give a general characterization for such processes. Then we prove that if all martingales of the basis B 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 F-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 (F(t)). Finally we prove a functional convergence result toward a process X of this kind.