Existence of densities for jumping stochastic differential equations

We consider a jumping Markov process { X t x } t ≥ 0 . We study the absolute continuity of the law of X t x for t > 0 . We first consider, as Bichteler and Jacod [K. Bichteler, J. Jacod, Calcul de Malliavin pour les diffusions avec sauts, existence d’une densité dans le cas unidimensionel, in: Séminaire de Probabilités XVII, in: L.N.M., vol. 986, Springer, 1983, pp. 132–157] did, the case where the rate of jumping is constant. We state some results in the spirit of those of [K. Bichteler, J. Jacod, Calcul de Malliavin pour les diffusions avec sauts, existence d’une densité dans le cas unidimensionel, in: Séminaire de Probabilités XVII, in: L.N.M., vol. 986, Springer, 1983, pp. 132–157], with rather weaker assumptions and simpler proofs, not relying on the use of stochastic calculus of variations. We next extend our method to the case where the rate of jumping depends on the spatial variable, and this last result seems to be new.