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

Zygalakis, K. C.

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

Type

Article

Authors

Zygalakis, K. C.

KAUST Grant Number

KUK-C1-013-04

Date

2011-01

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.

Citation

Zygalakis KC (2011) On the Existence and the Applications of Modified Equations for Stochastic Differential Equations. SIAM Journal on Scientific Computing 33: 102–130. Available: http://dx.doi.org/10.1137/090762336.

Acknowledgements

This work was partially supported by award KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). It was also partially funded by a David Crighton Fellowship.