No 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 .

Abstract. The density of real-valued L evy processes is studied in small time under the assumption that the process has many small jumps. We prove that the real line can be divided into three subsets on which the density is smaller and smaller: the set of points that the process can reach with a nite number of jumps ( -accessible points) the set of points that the process can reach with an in nite number of jumps (asymptotically -accessible points) and the set of points that the process cannot reach b y jumping ( -inaccessible points).