Malliavin calculus for stochastic differential equations with jumps

第一个结果是关于一类跳跃过程系统的滤波方程解的正则性。利用Malliavin微积分中的一个新的关键引理，在广义Hormander条件下证明了解的光滑稠密性。第二个结果是：设a_0(x，x^^-)是R^N×R^N上的R^N值光滑函数，v(dθ)是离散空间上的有限测度，θ，β_t=(β^θ_t)是维纳过程，μ(DV)是满足强可积条件的测度。考虑一类特殊类型的随机微分方程组：对u∈R^d，[数值公式]假设x^u(0)在u中光滑，过程x(T)=(x^u(T))取值于Hilbert空间H=(L^2(R^d，B(R^d)，μ))^N，设π：H→R^M是有界线性映射.随机变量定律π(x(T))的光滑密度的存在称为随机变量的偏亚椭圆性。在H上引入向量场的偏Hormander条件：[数值公式]借助于一个新的Key引理，证明了在偏Hormander条件下偏亚椭圆性所满足的偏Hormander定理，并对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一SDE系统定义的随机流诱导的测度的绝对连续性的传播问题。

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.