An Asymptotic Version of the Parametrix Method for Markov Chains Converging to Diffusions

Abstract

Abstract The paper provides a generalization of the local limit theorem on the convergence of inhomogeneous Markov chains to the diffusion limit for the case in which the corresponding coefficients of the process satisfy weak regularity conditions and coincide only asymptotically. In particular, the drift coefficients under consideration can be unbounded with at most linear growth, and the bounds reflect the transfer of the terminal state by an unbounded trend through the corresponding deterministic flow. Our approach is based on studying the uniform distance between transition densities of a given inhomogeneous Markov chain and the limit diffusion process, and the bound for the convergence rate was obtained using the classical local limit theorem and the stability bounds of the parametrix type.