Smooth densities for solutions to stochastic differential equations with jumps

We consider a solution xtxt to a generic Markovian jump diffusion and show that for any t0>0t0>0 the law of xt0xt0 has a C∞C∞ density with respect to the Lebesgue measure under a uniform version of the Hörmander conditions. Unlike previous results in the area the result covers a class of infinite activity jump processes. The result is accomplished using carefully crafted refinements to the classical arguments used in proving the smoothness of density via Malliavin calculus. In particular, we provide a proof that the semimartingale inequality of J. Norris persists for discontinuous semimartingales when the jumps are small.