High order steady-state diffusion approximations

TL;DR

This paper introduces higher-order diffusion approximations for stationary distributions of Markov chains, improving accuracy over traditional methods while maintaining similar computational complexity, supported by theoretical and numerical evidence.

Contribution

It develops recursive, higher-order diffusion approximations using Stein's method, advancing the accuracy of stationary distribution estimates for Markov chains.

Findings

Higher-order approximations outperform classical diffusion methods in accuracy.

The approach maintains computational complexity comparable to traditional methods.

Theoretical and numerical results validate the effectiveness of the new approximations.

Abstract

We derive and analyze new diffusion approximations of stationary distributions of Markov chains that are based on second- and higher-order terms in the expansion of the Markov chain generator. Our approximations achieve a higher degree of accuracy compared to diffusion approximations widely used for the past fifty years, while retaining a similar computational complexity. To support our approximations, we present a combination of theoretical and numerical results across three different models. Our approximations are derived recursively through Stein/Poisson equations, and the theoretical results are proved using Stein's method.