WIAS Preprint No. 727, (2002)

Numerical methods for Langevin type equations based on symplectic integrators


  • Milstein, Grigori N.
  • Tretyakov, Michael V.

2010 Mathematics Subject Classification

  • 65C30 65P10 82C31


  • Langevin equations, stochastic Hamiltonian systems, symplectic and quasi-symplectic numerical methods, mean-square and weak schemes


Langevin type equations are an important and fairly large class of systems close to Hamiltonian ones. The constructed mean-square and weak quasi-symplectic methods for such systems degenerate to symplectic methods when a system degenerates to a stochastic Hamiltonian one. In addition, quasi-symplectic methods' law of phase volume contractivity is close to the exact law. The methods derived are based on symplectic schemes for stochastic Hamiltonian systems. Mean-square symplectic methods were obtained in citehadd,hmul while symplectic methods in the weak sense are constructed in this paper. Special attention is paid to Hamiltonian systems with separable Hamiltonians, with additive noise, and with colored noise. Some numerical tests of both symplectic and quasi-symplectic methods are presented. They demonstrate superiority of the proposed methods in comparison with standard ones.

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