Discrete approximations to reflected Brownian motion

Details Discrete approximations to reflected Brownian motion Discrete approximations to reflected Brownian motion

2006-11-05

arxiv.org/abs/math/0611114v2

Annals of Probability 2008, Vol. 36, No. 2, 698-727

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/009117907000000240 the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

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 $\mathbb{R}^n$ 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 $D\cap2^{-k}\mathbb{Z}^n$ moving at the rate $2^{-2k}$ with stationary initial distribution converge weakly in the space $\mathbf{D}([0,1],\mathbb{R}^n)$, 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 $k\geq1$, 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.

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