Discrete approximations to reflected Brownian motion

Abstract

In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains D in ℝⁿ that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on D∩2^−kℤⁿ moving at the rate 2^−2k with stationary initial distribution converge weakly in the space D([0, 1], ℝⁿ), equipped with the Skorokhod topology, to the law of the stationary reflected Brownian motion on D. We further show that the following “myopic conditioning” algorithm generates, in the limit, a reflected Brownian motion on any bounded domain D. For every integer k≥1, let {X^k_j2^−k, j=0, 1, 2, …} be a discrete time Markov chain with one-step transition probabilities being the same as those for the Brownian motion in D conditioned not to exit D before time 2^−k. We prove that the laws of X^k converge to that of the reflected Brownian motion on D. These approximation schemes give not only new ways of constructing reflected Brownian motion but also implementable algorithms to simulate reflected Brownian motion.