Using moment approximations to study the density of jump driven SDEs

In order to study the regularity of density a solution infinite activity jump driven stochastic differential equation we consider following two-step approximation method. First, use moment problem in approximate small jumps by another whose Lévy measure has finite support. second step replace first two moments noise Brownian motion based on Assmussen-Rosiński approach. This needs satisfy certain properties apply “balance” method which allows densities for process Malliavin Calculus motion. Our results situations where is absolutely continuous with respect Lebesgue or purely atomic measures combinations them.