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Efficient Stochastic Runge-Kutta Methods for Stochastic Differential Equations with Small Noises

Efficient Stochastic Runge-Kutta Methods for Stochastic Differential Equations with Small Noises
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Year:    2018

Author:    Xiao Tang, Aiguo Xiao

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 4 : pp. 845–878

Abstract

New stochastic Runge-Kutta (SRK) methods for solving the Itô and Stratonovich stochastic differential equations (SDEs) with small noises are introduced. These SRK methods contain some multiple stochastic integrals simulated easily and have high global mean-square error accuracy. To simplify the calculation process, the stochastic rooted tree analysis is developed to estimate the local error and the global mean-square error estimate for a general class of SRK methods is given. Various improved SRK methods for the Itô or Stratonovich SDEs with non-commutative, commutative, diagonal, scalar, additive or colored small noises are proposed in turn. Finally, the proposed new SRK methods are examined by four test equations and all of the numerical results show the high efficiency of our methods.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2017-0181

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 4 : pp. 845–878

Published online:    2018-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    34

Keywords:    Stochastic differential equations stochastic Runge-Kutta methods small noises.

Author Details

Xiao Tang

Aiguo Xiao

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