Generalized Hormander theorem for non-local operators

The smoothness of marginal distributions of solutions to SDE's with jumps is investigated via the Malliavin calculus on the cad-lag space. A key lemma on the estimation of some functionals of semi-martingales is presented. Using the key lemma, the Hormander theorem on the hypo-ellipticity of parabolic differential operators is generalized to a theorem for the integro-differential operators associated with the SDE's. This article is essentially a summary of [3]. But the last section where the main theorem is presented is an improvement of [3].