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Volume 43, Issue 3
Central Limit Theorem for Temporal Average of Backward Euler-Maruyama Method

Diancong Jin

DOI: 10.4208/jcm.2311-m2023-0147

J. Comp. Math., 43 (2025), pp. 588-614.

Published online: 2024-11

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  • 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 (SODEs). We give the central limit theorem (CLT) of the temporal average of 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 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. The main contribution of this work is to generalize the existing CLT of the temporal average of numerical methods to that for SODEs with super-linearly growing drift coefficients.

  • Keywords

Central limit theorem, Temporal average, Ergodicity, Backward Euler-Maruyama method.

  • AMS Subject Headings

60H35, 60F05, 60H10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-43-588, author = {Jin , Diancong}, title = {Central Limit Theorem for Temporal Average of Backward Euler-Maruyama Method}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {43}, number = {3}, pages = {588--614}, 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 (SODEs). We give the central limit theorem (CLT) of the temporal average of 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 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. The main contribution of this work is to generalize the existing CLT of the temporal average of numerical methods to that for SODEs with super-linearly growing drift coefficients.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2311-m2023-0147}, url = {http://global-sci.org/intro/article_detail/jcm/23551.html} }
TY - JOUR T1 - Central Limit Theorem for Temporal Average of Backward Euler-Maruyama Method AU - Jin , Diancong JO - Journal of Computational Mathematics VL - 3 SP - 588 EP - 614 PY - 2024 DA - 2024/11 SN - 43 DO - http://doi.org/10.4208/jcm.2311-m2023-0147 UR - https://global-sci.org/intro/article_detail/jcm/23551.html KW - Central limit theorem, Temporal average, Ergodicity, Backward Euler-Maruyama method. AB -

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 (SODEs). We give the central limit theorem (CLT) of the temporal average of 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 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. The main contribution of this work is to generalize the existing CLT of the temporal average of numerical methods to that for SODEs with super-linearly growing drift coefficients.

Jin , Diancong. (2024). Central Limit Theorem for Temporal Average of Backward Euler-Maruyama Method. Journal of Computational Mathematics. 43 (3). 588-614. doi:10.4208/jcm.2311-m2023-0147
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