The simulation of phase trajectories of a diffusion process in a bounded domain

Abstract

As distinct from the usual approximation of solutions of stochastic differential equations (SDE) when the time-discretization (in other words sampling) is exploited, here a space-discretization (in other words quantization) for approximation of phase trajectories of autonomous systems of SDE is proposed. For systems with zero drift the next approximate point on the phase trajectory is found by a random walk over the boundary of a small ellipsoid with centre at the previous point. Theorems on mean-square order of accuracy for such an approximation are proved. An algorithm fur approximate construction of exit point from a bounded domain is given. In the t-case of a general system a surface on which it is possible effectively to simulate the next approximate point of the phase trajectory, is constructed