Stochastic calculus of variations for general Lévy processes and its applications to jump-type SDE's with non-degenerated drift

Abstract

We consider an SDE in R^m of the type dX(t)=a(X(t))dt+dU(t) with a Lévy process U and study the problem for the distribution of a solution to be regular in various senses. We do not impose any specific conditions on the Lévy measure of the noise, and this is the main difference between our method and the known methods by J.Bismut or J.Picard. The main tool in our approach is the stochastic calculus of variations for a Lévy process, based on the time-stretching transformations of the trajectories. Three problems are solved in this framework. First, we prove that if the drift coefficient a is non-degenerated in an appropriate sense, then the law of the solution to the Cauchy problem for the initial equation is absolutely continuous, as soon as the Lévy measure of the noise satisfies one of the rather weak intensity conditions, for instance the so-called wide cone condition. Secondly, we provide the sufficient conditions for the density of the distribution of the solution to the Cauchy problem to be smooth in the terms of the family of the so-called order indices of the Lévy measure of the noise (the drift again is supposed to be non-degenerated). At last, we show that an invariant distribution to the initial equation, if exists, possesses a C^\infty-density provided the drift is non-degenerated and the Lévy measure of the noise satisfies the wide cone condition.