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

K. C. Zygalakis

doi:10.1137/090762336

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

K. C. Zygalakis

Research output: Contribution to journal › Article › peer-review

39 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 language	English (US)
Pages (from-to)	102-130
Number of pages	29
Journal	SIAM Journal on Scientific Computing
Volume	33
Issue number	1
DOIs	https://doi.org/10.1137/090762336
State	Published - Jan 2011
Externally published	Yes

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title = "On the Existence and the Applications of Modified Equations for Stochastic Differential Equations",

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. {\textcopyright} 2011 Society for Industrial and Applied Mathematics.",

author = "Zygalakis, {K. C.}",

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year = "2011",

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N2 - 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.

AB - 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.

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

Abstract

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