Study on regularities of stochastic processes with jumps

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We studied on the Hormander theorem via the Malliavin calculus for the parabolic pseudo-differential operator :<<numerical formula>>where <<numerical formula>>, and B_θ are operators defined by <<numerical formula>>. The hypoellipticity of the above operator is equivalent to the existence of smooth densities of transition probabilities of Markov processes with jumps which are solutions to stochastic integro-differential equations :<<numerical formula>>where β(s)=(β^κ(s)) is an m-dimensional Brownian motion and J(dsdθ) is a Poisson random measure with E[J(dsdθ)]=π(dθ)ds. A similar problem for usual parabolic differential operators was studied in the course of the Malliavin calculus.We carried out the variation for jump type Markov processes by Girzanov transforms of Levy processes, and proved special necessary formulas of integration by parts on the cad-lag space. It must be shown the strong decay of the Laplace transform of the distribution of a specific functional associated with the Malliavin covariance. So far, similar strong decay property was proved by long complicated arguments. But we proved it by a new simple method where the key lemma is an estimate for general semimartingales. We proved the smoothness of transition densities of Markov processes with jumps under certain conditions which are essentially weaker than the Hormander type condition introduced by Leandre.