The martingale problem for generators of variable order

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We studied on the existence of smooth densities of transition probabilities of Markov processes with jumps which are solutions to d-dimensional stochastic integro-differential equations:dxィイD2tィエD2 = aィイD20ィエD2(xィイD2tィエD2)dt + ΣィイD6m(/)k=1ィエD6aィイD2kィエD2(xィイD2tィエD2)・dβィイD1kィエD1(t) + ∫b(xィイD2t-ィエD2,θ)J(dtdθ),where β(t) = (βィイD1kィエD1(t)) is an m-dimensional Brownian motion and J(dtdθ) is a Poisson random measure with E[J(dtdθ)] = π(dθ)dt. The existence of smooth densities is equivalent to the hypoellipticity of the parabolic pseudo-differential operatorィイD7∂(/)∂tィエD7 + (AィイD20ィエD2 + ィイD71(/)2ィエD7ΣィイD6m(/)k=1ィエD6(AィイD2kィエD2)ィイD12ィエD1 + ∫(BィイD2θィエD2 - I)π(dθ)),where AィイD2kィエD2 = aィイD2kィエD2(x)・∂ィイD2xィエD2 and BィイD2θィエD2 are operators defined by BィイD2θィエD2φ(x) = φ(x + b(x,θ)).A similar problem for continuous Markov processes was studied in the course of the Malliavin calculus, and the Hormander condition is well-known as a sufficient condition for the hypoellipticity. 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. We also faced the problem to show the exponential decay of the Laplace transform of the distribution of a specific functional associated with the Malliavin covariance. So far, the exponential decay was proved by long complicated arguments. We showed it by a new method where the key lemma is an estimate for general semimartingales. And we proved the smoothness of transition densities of Markov processes with jumps under certain conditions which are weaker than Hormander type conditions in the previous sense. This method also gives a quite simple proof to the Hormander theorem for usual parabolic differential operators.