WIAS Preprint No. 670, (2001)

Mean-square symplectic methods for Hamiltonian systems with multiplicative noise


  • Milstein, Grigori N.
  • Repin, Yuri M.
  • Tretyakov, Michael V.

2010 Mathematics Subject Classification

  • 60H10 65U05


  • Stochastic Hamiltonian systems, symplectic integration, implicit methods, mean-square convergence


Stochastic systems with multiplicative noise, phase flows of which have integral invariants, are considered. For such systems, numerical methods preserving the integral invariants are constructed using full implicit schemes of a new type for stochastic differential equations. In these full implicit schemes increments of Wiener processes are substituted by some truncated random variables. They are important for both theory and practice of numerical integration of stochastic differential equations. A special attention is paid to systems with separable Hamiltonians and to Hamiltonian systems with small noise. Liouvillian methods for stochastic systems preserving phase volume are also proposed. Some results of numerical experiments are presented.

Appeared in

  • SIAM J. on Numerical Analysis, vol. 40 (2003), no. 4, pp. 1583-1604, under new title: Numerical methods for stochastic systems preserving symplectic structure.

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