Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump

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.