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We consider a one-dimensional jumping Markov process $\{X^x_t\}_{t \geq 0}$,solving a Poisson-driven stochastic differential equation. We prove that thelaw of $X^x_t$ admits a smooth density for $t>0$, under some regularity andnon-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 rateof 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 calculustechniques. To overcome this problem, we introduce a new method, in which thepropagation of the smoothness of the density is obtained by analytic arguments.

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