Study on regularities of stochastic processes with jumps

利用Malliavin演算研究了抛物型伪微分算子的Hormander定理：<<数值公式>>，其中<<数值公式>>，和B_θ是由<<数值公式>>定义的算子。上述算子的亚椭圆性等价于具有跳跃的马尔可夫过程的转移概率的光滑密度的存在性，这些马尔可夫过程是随机积分微分方程的解：<<数值公式>>，其中β(s)=(β^κ(s))是m维布朗运动，J（dsdθ）是E[J(dsdθ)]=π(dθ)ds的泊松随机测度。在马利文微积分课程中研究了一般抛物型微分算子的一个类似问题。利用Levy过程的Girzanov变换对跳跃型马尔可夫过程进行变分，并证明了cadlag空间上部分积分的特殊必要公式。必须证明与马利文协方差相关的特定泛函分布的拉普拉斯变换的强衰减。到目前为止，类似的强衰变性质是通过冗长复杂的论证证明的。但我们用一种新的简单方法证明了它，其中关键引理是对一般半鞅的估计。我们证明了具有跳跃的马尔可夫过程在某些条件下转移密度的平滑性，这些条件实质上弱于Leandre引入的Hormander型条件。

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.