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

TL;DR

This paper proves a truncated local limit theorem for multidimensional diffusion approximations, showing their transition probabilities converge uniformly to the diffusion density using a novel Malliavin calculus approach.

Contribution

It introduces a new Malliavin calculus framework for measures with singular components, enabling analysis of difference approximations of multidimensional diffusions.

Findings

Transition densities of approximations converge uniformly to the diffusion density.

Provides uniform estimates for mixing and convergence rates of difference schemes.

Establishes convergence results for local times of multidimensional diffusions.

Abstract

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