A randomized quasi-Monte Carlo simulation method for Markov chains
成果类型：
Article
署名作者：
L'Ecuyer, Pierre; Lecot, Christian; Tuffin, Bruno
署名单位：
Universite de Montreal; Universite de Montreal; Universite Savoie Mont Blanc; Universite Paris Saclay; Universite de Rennes
刊物名称：
OPERATIONS RESEARCH
ISSN/ISSBN：
0030-364X
DOI：
10.1287/opre.1080.0556
发表日期：
2008
页码：
958-975
关键词：
摘要：
We introduce and study a randomized quasi-Monte Carlo method for the simulation of Markov chains up to a random (and possibly unbounded) stopping time. The method simulates n copies of the chain in parallel, using a (d + 1)-dimensional, highly uniform point set of cardinality n, randomized independently at each step, where d is the number of uniform random numbers required at each transition of the Markov chain. The general idea is to obtain a better approximation of the state distribution, at each step of the chain, than with standard Monte Carlo. The technique can be used in particular to obtain a low-variance unbiased estimator of the expected total cost when state-dependent costs are paid at each step. It is generally more effective when the state space has a natural order related to the cost function. We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial. The variance can be reduced by factors of several thousands in some cases. We prove bounds on the convergence rate of the worst-case error and of the variance for special situations where the state space of the chain is a subset of the real numbers. In line with what is typically observed in randomized quasi-Monte Carlo contexts, our empirical results indicate much better convergence than what these bounds guarantee.