WIAS Preprint No. 123, (1994)

Weak approximation for stochastic differential equations with small noises


  • Milstein, Grigori N.
  • Tret´yakov, Michael V.

2010 Mathematics Subject Classification

  • 60H10


  • Stochastic differential equations, small noises, numerical methods, Monte-Carlo technique


New approach to construction of weak numerical methods, which are intended for Monte-Carlo technique, is proposed for a stochastic system with small noises. The theorem on estimate of method error in terms of product hiε j (h is a time increment, ε is a small parameter) is proved. Various efficient weak schemes are derived for a general system with small noises and for systems with small additive and small colored noises. The Talay-Tubaro expansion of the global error is considered for such systems. The efficient approach to reduction of the Monte-Carlo error is proposed. The derived methods are tested by calculation of Lyapunov exponents and by simulation of a bistable dynamical system for which multiplicative stochastic resonance is observed.

Appeared in

  • SIAM J. on Numerical Analysis, vol. 34 (1997), no. 6, pp. 2142-2167, under new title: Numerical methods in the weak sense for stochastic differential equations with small noise.

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