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A Numerical Scheme for Invariant Distributions of Constrained Diffusions

Amarjit Budhiraja 1, Jiang Chen 1, Sylvain Rubenthaler 2

(10/09/2011)

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

1 :	Department of Statistics and Operations Research, University of North Carolina
	UNC Chapel Hill
2 :	Laboratoire Jean Alexandre Dieudonné (JAD)
	CNRS : UMR6621 – Université de Nice Sophia Antipolis (UNS)

Équipe de recherche : Probabilités et Statistiques, Laboratoire J. A. Dieudonné, université de Nice-Sophia Antipolis & Department of Statistics and Operations Research, University of North Carolina at Chapel Hill

Domaine

:

Mathématiques/Probabilités

Mots Clés : Reflected Diffusions – Heavy Traffic Theory – Stochastic Networks – Skorohod Problem – Invariant Measures – Stochastic Algorithms.

hal-00622153, version 1

http://hal-unice.archives-ouvertes.fr/hal-00622153

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Contributeur : Sylvain Rubenthaler <>

Soumis le : Lundi 12 Septembre 2011, 09:55:08

Dernière modification le : Vendredi 16 Septembre 2011, 17:55:19