Markov chain approximations for transition densities of L\'evy processes

TL;DR

This paper analyzes how well a lattice-based Markov chain approximates the transition densities of Levy processes, providing convergence rates in various dimensions under specific conditions.

Contribution

It establishes sharp convergence rates for Markov chain approximations of Levy processes, extending results to higher dimensions with technical conditions.

Findings

Convergence rates are sharp in one dimension.

Higher-dimensional convergence depends on non-degeneracy of the diffusion matrix.

Provides conditions under which transition densities exist and are approximated accurately.

Abstract

We consider the convergence of a continuous-time Markov chain approximation X^h, h>0, to an R^d-valued Levy process X. The state space of X^h is an equidistant lattice and its Q-matrix is chosen to approximate the generator of X. In dimension one (d=1), and then under a general sufficient condition for the existence of transition densities of X, we establish sharp convergence rates of the normalised probability mass function of X^h to the probability density function of X. In higher dimensions (d>1), rates of convergence are obtained under a technical condition, which is satisfied when the diffusion matrix is non-degenerate.