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

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2010-07-07

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Mattingly, Jonathan Christopher

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Abstract

Numerical approximation of the long time behavior of a stochastic di.erential 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. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

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Published Version (Please cite this version)

10.1137/090770527

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Mattingly, Jonathan Christopher, AM Stuart and MV Tretyakov (2010). Convergence of numerical time-averaging and stationary measures via Poisson equations. SIAM Journal on Numerical Analysis, 48(2). pp. 552–577. 10.1137/090770527 Retrieved from https://hdl.handle.net/10161/4314.

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