Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations

TL;DR

This paper establishes error estimates for numerical time-averaging of SDEs, demonstrating convergence of the numerical stationary distribution to the true distribution using a Poisson equation approach, applicable to various schemes.

Contribution

It introduces a simple, universal method based on Poisson equations for analyzing long-term behavior of numerical SDE approximations, including hypoelliptic cases.

Findings

Error estimates for time-averaging estimators

Convergence of numerical stationary measures to true measures

Applicable to explicit and implicit schemes, including hypoelliptic SDEs

Abstract

Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantage of this approach is its simplicity and universality. It works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus and we study only smooth test functions. However we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed.