WIAS Preprint No. 352, (1997)

Space-time random walk for stochastic differential equations in a bounded domain


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

2010 Mathematics Subject Classification

  • 60H10 60J15 65U05


  • space-time Brownian motion, mean-square approximation, the Dirichlet problem for equations of parabolic and elliptic type, exit points




A mean-square approximation, which ensures boundedness of both time and space increments, is considered for stochastic differential equations in a bounded domain. The proposed algorithm is based on a space-time discretization using a random walk over boundaries of small space-time parallelepipeds. To realize the algorithm, exact distributions for exit points of the space-time Brownian motion from a space-time parallelepiped are given. Convergence theorems are stated for the proposed algorithm. A method of approximate searching for exit points of the space-time diffusion from the bounded domain is constructed. Results of several numerical tests are presented.

Appeared in

  • Annals of Applied Probability, vol. 9 (1999), no.3, pp. 732-779, under new title: Simulation of a space-time bounded diffusion.

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