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

TL;DR

The article advances the numerical analysis of ergodic SDEs with super-linear drift by establishing central limit theorems for the temporal average of the backward Euler–Maruyama discretization. It handles two deviation regimes: for $\alpha\in(1,2)$, CLTs follow from uniform strong convergence; for $\alpha=2$, a Poisson-equation decomposition is employed to derive the same normal limit, with $\varphi$ solving $\mathcal{L}\varphi=h-\pi(h)$. The authors also prove $p$th moment bounds for the BEM in the infinite horizon and provide numerical experiments that corroborate the theory. The results extend CLTs for numerical temporal averages to SDEs with non-Lipschitz, super-linear drift, offering rigorous justification for long-time statistical approximations in challenging settings.

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.

Figures (2)

• Figure 1: The expectations of $h(\bar{X}_n)$ starting from different initial values with $\tau=2^{-14}$, $M=5000$.
• Figure 2: The expectations of $(\bar{X}_n)^5$ starting from different initial values with $\tau=2^{-14}$.

Theorems & Definitions (21)

• Remark 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 3.1
• Theorem 3.2
• proof
• Remark 3.3
• Theorem 3.4
• Theorem 4.1