Malliavin calculus for stochastic differential equations with jumps

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The first result is on the regularity of solution to the filtering equation for a certain system of jump type processes. Applying a new key lemma in the Malliavin calculus, the existence of smooth density of the solution was proved under a generalized Hormander condition. The second result is the following :Let a_0(x,x^^-) be an R^N - valued smooth function on R^N × R^N , v(dθ) be a finite measure on a discrete space θ, β_t = (β^θ_t) be a Wiener process, and let μ(dv) be a measure satisfying a strong integrability condition. Consider a system of SDE's of the specific type : for u ∈ R^d, 【numerical formula】Assume that x^u(0) is smooth in u and the process x(t) = (x^u(t)) takes its values in the Hilbert space H = (L^2(R^d,B(R^d),μ))^N. Let π: H → R^M be a bounded linear mapping. The existence of the smooth density of the law of the random variable π(x(T)) is called the partial hypoellipticity of the SDE. Introduce the partial Hormander condition for vector fields on H : 【numerical formula】The partial Hormander theorem that the partial hypoellipticity holds under the partial Hormander condition is proved proceeding the Malliavin calculus for SDE's on the Hilbert space H with the help of a new key lemma. The partial Hormander theorem can be applied to the problem about the propagation of absolute continuity of measures induced by stochastic flows defined by a certain system of SDE's.