On the Existence and the Applications of Modified Equations for Stochastic Differential Equations

K. C. Zygalakis

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27 Scopus citations

Abstract

In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive an SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations is also discussed. © 2011 Society for Industrial and Applied Mathematics.
Original languageEnglish (US)
Pages (from-to)102-130
Number of pages29
JournalSIAM Journal on Scientific Computing
Volume33
Issue number1
DOIs
StatePublished - Jan 2011
Externally publishedYes

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