Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem

Abstract

For difference approximations of multidimensional diffusions, the truncated local limit theorem is proved. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities that converge uniformly to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain non-trivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times of multidimensional diffusions are given.