Nondegenerate SDE's with jumps and their hypoelliptic properties

Abstract

We study ‘nondegenerate’ SDE's with jumps. These include SDE satisfying ‘point-wise positive’ condition and that satisfying (nonstationary) Hörmander's condition. We show that solutions of these SDE's have hypoelliptic properties. Our result is based on the Malliavin calculus on the Wiener-Poisson space. In case of continuous SDE, it extends and refines works based on the Malliavin calculus on the Wiener space.