A Numerical Scheme for Invariant Distributions of Constrained Diffusions

• 成果类型: Article
• 署名作者: Budhiraja, Amarjit; Chen, Jiang; Rubenthaler, Sylvain
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; Universite Cote d'Azur; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2013.0599
• 发表日期: 2014
• 页码: 262-289
• 关键词: reflecting brownian motions; queuing-networks; Sufficient conditions; ergodic control; approximations; EXISTENCE; orthant

摘要

Reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic. Stationary distributions of such models give useful information on the steady-state performance of the corresponding stochastic networks, and thus it is important to develop reliable and efficient algorithms for numerical computation of such distributions. In this work we propose and analyze a Monte-Carlo scheme based on an Euler type discretization of the reflected stochastic differential equation using a single sequence of time discretization steps which decrease to zero as time approaches infinity. Appropriately weighted empirical measures constructed from the simulated discretized reflected diffusion are proposed as approximations for the invariant probability measure of the true diffusion model. Almost sure consistency results are established That, in particular, show that weighted averages of polynomially growing continuous functionals evaluated on the discretized simulated system converge a.s. to the corresponding integrals with respect to the invariant measure. Proofs rely on constructing suitable Lyapunov functions for tightness and uniform integrability and characterizing almost sure limit points through an extension of Echeverria's criteria for reflected diffusions. Regularity properties of the underlying Skorohod problems play a key role in the proofs. Rates of convergence for suitable families of test functions are also obtained. A key advantage of Monte-Carlo methods is the ease of implementation, particularly for high-dimensional problems. A numerical example of an eight-dimensional Skorohod problem is presented to illustrate the applicability of the approach.