Central Limit Theorem for Temporal Average of Backward Euler-Maruyama Method (Journal of Computational Mathematics, 2024)
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 differentialequations (SODEs). We give the central limit theorem (CLT) of the temporal averageof the BEM method, which characterizes its asymptotics in distribution. When the deviation order is smaller than the optimal strong order, we directly derive the CLT of thetemporal average through that of original equations and the uniform strong order of theBEM 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 originalequations. Numerical experiments are performed to illustrate the theoretical results. Themain contribution of this work is to generalize the existing CLT of the temporal averageof numerical methods to that for SODEs with super-linearly growing drift coefficients.