The Annals of Probability

Discrete approximations to reflected Brownian motion

Krzysztof Burdzy and Zhen-Qing Chen

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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∩2kn 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 {Xkj2k, 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 2k. We prove that the laws of Xk 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.

Article information

Ann. Probab., Volume 36, Number 2 (2008), 698-727.

First available in Project Euclid: 29 February 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60J60: Diffusion processes [See also 58J65] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 31C25: Dirichlet spaces

Reflected Brownian motion random walk killed Brownian motion conditioning martingale tightness Skorokhod space Dirichlet form


Burdzy, Krzysztof; Chen, Zhen-Qing. Discrete approximations to reflected Brownian motion. Ann. Probab. 36 (2008), no. 2, 698--727. doi:10.1214/009117907000000240.

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