Mean-square approximations, which ensure boundedness of both time and space increments, are constructed for stochastic differential equations in a bounded domain. The proposed algorithms are based on a space-time discretization using a random walk over boundaries of small space-time parallelepipeds. To realize the algorithms, 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 algorithms. 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.
"Simulation of a space-time bounded diffusion." Ann. Appl. Probab. 9 (3) 732 - 779, August 1999. https://doi.org/10.1214/aoap/1029962812