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March 2008 Discrete approximations to reflected Brownian motion
Krzysztof Burdzy, Zhen-Qing Chen
Ann. Probab. 36(2): 698-727 (March 2008). DOI: 10.1214/009117907000000240


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.


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Krzysztof Burdzy. Zhen-Qing Chen. "Discrete approximations to reflected Brownian motion." Ann. Probab. 36 (2) 698 - 727, March 2008.


Published: March 2008
First available in Project Euclid: 29 February 2008

zbMATH: 1141.60014
MathSciNet: MR2393994
Digital Object Identifier: 10.1214/009117907000000240

Primary: 60F17
Secondary: 31C25 , 60J10 , 60J60

Keywords: Conditioning , Dirichlet form , killed Brownian motion , martingale , Random walk , reflected Brownian motion , Skorokhod space , tightness

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 2 • March 2008
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