Central limit theorem for temporal average of backward Euler--Maruyama method

TL;DR

This paper establishes a central limit theorem for the temporal average of the backward Euler--Maruyama method applied to stochastic differential equations with super-linear drift, providing theoretical insights and numerical validation.

Contribution

It introduces the first CLT for the temporal average of BEM applied to ergodic SDEs with super-linear drift, covering different deviation orders.

Findings

CLT derived for deviation orders below the optimal strong order.

CLT established at the optimal strong order using Poisson equations.

Numerical experiments confirm theoretical results.

Abstract

This work focuses on the temporal average of the backward Euler--Maruyama (BEM) method, which is used to approximate the ergodic limit of stochastic ordinary differential equations with super-linearly growing drift coefficients. We give the central limit theorem (CLT) of the temporal average, which characterizes the asymptotics in distribution of the temporal average. When the deviation order is smaller than the optimal strong order, we directly derive the CLT of the temporal average through that of original equations and the uniform strong order of the BEM method. For the case that the deviation order equals to the optimal strong order, the CLT is established via the Poisson equation associated with the generator of original equations. Numerical experiments are performed to illustrate the theoretical results.