一类具有变跳率的一维跳变微分方程的光滑性
Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump
摘要：
考虑一维跳马尔可夫过程$\{X^x_t\}_{t \geq 0}$，求解泊松驱动的随机微分方程.证明了$X^x_t$定律允许$t>0$在S.D.E.系数的正则性和非退化性假设下有光滑密度.据我们所知，我们的结果是第一个包括一个非常数的跳跃率的重要情况。主要的困难在于，在这种情况下，映射$x \mapsto X^x_t$不平滑。这似乎使得不可能使用Malliavin演算技术。为了克服这个问题，我们引入了一种新的方法，在该方法中，密度的光滑性的传播是由解析参数得到的。
英文摘要：
We consider a one-dimensional jumping Markov process $\{X^x_t\}_{t \geq 0}$, solving a Poisson-driven stochastic differential equation. We prove that the law of $X^x_t$ admits a smooth density for $t>0$, under some regularity and non-degeneracy assumptions on the coefficients of the S.D.E. To our knowledge, our result is the first one including the important case of a non-constant rate of jump. The main difficulty is that in such a case, the map $x \mapsto X^x_t$ is not smooth. This seems to make impossible the use of Malliavin calculus techniques. To overcome this problem, we introduce a new method, in which the propagation of the smoothness of the density is obtained by analytic arguments.
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