Convergence rate of Euler--Maruyama scheme to the invariant probability measure under total variation distance for the SDEs

Abstract

This article shows the geometric decay rate of Euler-Maruyama scheme for one-dimensional stochastic differential equation towards its invariant probability measure under total variation distance. Firstly, the existence and uniqueness of invariant probability measure and the uniform geometric ergodicity of the chain are studied through introduction of non-atomic Markov chains. Secondly, the equivalent conditions for uniform geometric ergodicity of the chain are discovered, by constructing a split Markov chain based on the original Euler-Maruyama scheme.

This is a peer-reviewed and improved version of the previous one (version 2). There is a change of authorship to the original manuscript (version 1). The structure of the paper, some statements and proofs have been improved, some typos are corrected. Current version: 21 pages. Previous version: 22 pages