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

Xiao Tang & Aiguo Xiao

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

Published online: 2018-06

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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.

  • Keywords

Stochastic differential equations, stochastic Runge-Kutta methods, small noises.

  • AMS Subject Headings

60H35, 60H10, 65L06, 65L20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-10-845, author = {}, title = {Efficient Stochastic Runge-Kutta Methods for Stochastic Differential Equations with Small Noises}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {4}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0181}, url = {http://global-sci.org/intro/article_detail/aamm/12499.html} }
TY - JOUR T1 - Efficient Stochastic Runge-Kutta Methods for Stochastic Differential Equations with Small Noises JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 845 EP - 878 PY - 2018 DA - 2018/06 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2017-0181 UR - https://global-sci.org/intro/aamm/12499.html KW - Stochastic differential equations, stochastic Runge-Kutta methods, small noises. AB -

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.

Xiao Tang & Aiguo Xiao. (2020). Efficient Stochastic Runge-Kutta Methods for Stochastic Differential Equations with Small Noises. Advances in Applied Mathematics and Mechanics. 10 (4). 845-878. doi:10.4208/aamm.OA-2017-0181
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