Malliavin calculus for stochastic differential equations with jumps

第一个结果是关于某跳跃型过程系统的滤波方程解的正则性。应用Malliavin微积分中的一个新的关键引理，证明了在广义Hormander条件下解的光滑密度的存在性。第二个结果是：设a_0（x,x^^-）是R^N × R^N上的一个R^N值光滑函数，v（dθ）是离散空间θ上的一个有限测度，β_t = (β^θ_t)是一个Wiener过程，μ(dv)是一个满足强可积条件的测度。考虑一个特殊类型的SDE系统：对于u∈R^d，【数值公式】假设x^u(0)在u中是光滑的，并且过程x(t) = (x^u(t))在Hilbert空间H = (L^2(R^d，B(R^d),μ))^N中取其值。设π: H→R^M是一个有界线性映射。随机变量π(x(T))的光滑密度律的存在性称为SDE的部分半椭圆性。引入H上向量场的偏Hormander条件：【数值公式】证明了在偏Hormander条件下偏半椭圆成立的偏Hormander定理，利用一个新的关键引理对Hilbert空间H上的SDE进行了Malliavin演算。部分Hormander定理可以应用于由某SDE系统定义的随机流引起的测度的绝对连续性传播问题。

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.