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

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