Using coupling methods to estimate sample quality for stochastic differential equations

TL;DR

This paper introduces a probabilistic coupling-based method to estimate the quality of numerical solutions for stochastic differential equations by bounding the distance between their invariant measures.

Contribution

It presents a novel approach using coupling methods to quantify the error in long-term statistical properties of SDE approximations.

Findings

Effective estimation of invariant measure distance using coupling methods.

Algorithms validated on low and high-dimensional examples.

Provides quantitative bounds for sample quality in stochastic simulations.

Abstract

A probabilistic approach for estimating sample qualities for stochastic differential equations is introduced in this paper. The aim is to provide a quantitative upper bound of the distance between the invariant probability measure of a stochastic differential equation and that of its numerical approximation. In order to extend estimates of finite time truncation error to infinite time, it is crucial to know the rate of contraction of the transition kernel of the SDE. We find that suitable numerical coupling methods can effectively estimate such rate of contraction, which gives the distance between two invariant probability measures. Our algorithms are tested with several low and high dimensional numerical examples.