  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.

Probability