Volume 10, Issue 4
Efficient Stochastic Runge-Kutta Methods for Stochastic Differential Equations with Small Noises

Xiao Tang and Aiguo Xiao

10.4208/aamm.OA-2017-0181

Adv. Appl. Math. Mech., 10 (2018), pp. 845-878.

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  • 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.

  • History

Published online: 2018-06

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