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

TL;DR

This paper establishes that the Euler-Maruyama scheme for 1D stochastic differential equations converges geometrically to its invariant measure in total variation distance, independent of step size, through Markov chain analysis.

Contribution

It introduces a novel analysis of convergence rates for Euler-Maruyama under total variation, showing independence from step size and linking ergodicity to Markov chain properties.

Findings

Convergence rate is geometric and independent of step size.

Invariant measure exists uniquely and the chain is uniformly ergodic.

Markov chain techniques are used to analyze convergence.

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. It turns out that this convergence rate is independent with the step size under total variation distance.