Commun. Comput. Phys., 16 (2014), pp. 169-200.

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

Jian Deng 1*, Cristina Anton 2, Yau Shu Wong 1

1 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada.
2 Department of Mathematics and Statistics, Grant MacEwan University, Edmonton, AB T5J 4S2, Canada.

Received 31 October 2012; Accepted (in revised version) 19 November 2013
Available online 10 April 2014


The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

AMS subject classifications: 60H10, 65C30, 65P10

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Key words: Stochastic Hamiltonian systems, symplectic integration, mean-square convergence, high-order schemes.

*Corresponding author.
Email: (J. Deng), (C. Anton), (Y. S. Wong)

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