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)|
|Number of pages||29|
|Journal||SIAM Journal on Scientific Computing|
|State||Published - Jan 2011|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.