On the existence of weak solutions for stochastic differential equations with driving martingales and random measures

Abstract

In the present work we give a generalization, on the one hand, of the main result of [1] on the existence of weak solutions (in the sense of joint solution-measure) for equations with driving martingales and random measures the coefficients of which depend on the solution-process X at each t∊R + only through X t– , and on the other hand, of the main results of [2] on the existence of weak solutions (in the strict sense) for equations the coefficients of which depend on the past of X but with driving semimartingales. Thus we prove existence of weak solutions (in the strict sense) for the equations with driving martingales and random measures coefficients of which depend in a predictable way on the process X. In the course of the proof the weak solution turns out as in [1] and [2] to be regular, that is satisfying the condition 7.4 or 7.5 of [2]