Upper bounds for the derivatives of the density associated to solutions of stochastic differential equations with jumps

Abstract

In this article, we prove the existence and regularity of densities associated to stochastic differential equations driven by a jump process with infinite activity. Furthermore, we provide explicit bounds for the density and its derivatives. The main argument is based on two ideas: (1) The use of the regularity of the Brownian motion approximation of small jumps of an infinite activity process (the so-called Asmussen-Rosiński approximation); (2) The improvement of the approximation procedure using a Taylor like expansion of the law based on the so-called Levi parametrix method.