Steady-state simulation of reflected Brownian motion and related stochastic networks

TL;DR

This paper introduces novel algorithms for unbiased steady-state expectation estimation of multidimensional reflected Brownian motion, with polynomial complexity under certain conditions, and employs wavelets for approximating Brownian motion.

Contribution

It develops the first algorithms for unbiased steady-state estimation of multidimensional reflected Brownian motion, including complexity analysis and wavelet-based approximation techniques.

Findings

Algorithms achieve unbiased estimates with polynomial expected termination time.

Wavelet methods provide deterministic uniform approximation of Brownian motion.

Methodology extends beyond steady-state simulation to other stochastic processes.

Abstract

This paper develops the first class of algorithms that enable unbiased estimation of steady-state expectations for multidimensional reflected Brownian motion. In order to explain our ideas, we first consider the case of compound Poisson (possibly Markov modulated) input. In this case, we analyze the complexity of our procedure as the dimension of the network increases and show that, under certain assumptions, the algorithm has polynomial-expected termination time. Our methodology includes procedures that are of interest beyond steady-state simulation and reflected processes. For instance, we use wavelets to construct a piecewise linear function that can be guaranteed to be within $\varepsilon$ distance (deterministic) in the uniform norm to Brownian motion in any compact time interval.