On the existence of smooth densities for jump processes

Summary We consider a Lévy processX _t and the solutionY _t of a stochastic differential equation driven byX _t; we suppose thatX _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 forY _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 variablesF defined on a Poisson probability space; the basic tool is a duality formula from which we estimate the characteristic function ofF.