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Convergence of numerical time-averaging and stationary measures via Poisson equations

Mattingly, J. C. and Stuart, A. M. and Tretyakov, M. (2010) Convergence of numerical time-averaging and stationary measures via Poisson equations. SIAM Journal of Numerical Analysis, 48 (2). pp. 552-577. ISSN 0036-1429. https://resolver.caltech.edu/CaltechAUTHORS:20160804-165401982

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Abstract

Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein’s method is indicated. Some practical implications of the results are discussed.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1137/090770527DOIArticle
http://epubs.siam.org/doi/abs/10.1137/090770527PublisherArticle
https://arxiv.org/abs/0908.4450v3arXivDiscussion Paper
Additional Information:© 2010 Society for Industrial and Applied Mathematics. Received by the editors September 9, 2009; accepted for publication (in revised form) March 5, 2010; published electronically May 21, 2010. [JCM's] work was supported by NSF grants DMS-0616710 and DMS-0449910 and a Sloan Fellowship. [AMS's] work was supported by the ERC and EPSRC. [MVT's] work was partially supported by the EPSRC research grant EP/D049792/1.
Funders:
Funding AgencyGrant Number
NSFDMS-0616710
NSFDMS-0449910
Alfred P. Sloan FoundationUNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)EP/D049792/1
Subject Keywords:stochastic differential equations, ergodic limits, convergence of weak schemes, time averaging, Poisson equation
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ83
Issue or Number:2
Classification Code:AMS subject classifications. Primary, 65C30; Secondary, 60H35, 37H10, 60H10
Record Number:CaltechAUTHORS:20160804-165401982
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160804-165401982
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69457
Collection:CaltechAUTHORS
Deposited By: Linda Taddeo
Deposited On:08 Aug 2016 17:05
Last Modified:03 Oct 2019 10:22

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