On the existence of smooth densities for jump processes

Abstract

We consider a Lévy process X t and the solution Y t of a stochastic differential equation driven by X t; we suppose that X t has infinitely many small jumps, but its Lévy measure may be very singular (for instance it may have a countable support). We obtain sufficient conditions ensuring the existence of a smooth density for Y t: these conditions are similar to those of the classical Malliavin calculus for continuous diffusions. More generally, we study the smoothness of the law of variables F defined on a Poisson probability space; the basic tool is a duality formula from which we estimate the characteristic function of F.