Jump-diffusion processes in random environments

In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions of SDEs. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate the Markov property. To prove uniqueness we solve a general martingale problem for càdlàg processes. This result is of independent interest. Application of our results to generalized exponential Lévy model are present in the last section.