A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

We introduce and study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain. The number of steps in the chain can be random and unbounded. 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. This technique is effective in particular to obtain a low-variance unbiased estimator of the expected total cost up to some random stopping time, 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-c...

• Type: Journal
• Year: 2008
• Where: IOR
• Authors: Pierre L'Ecuyer, Christian Lécot, Bruno Tuffin
• Keywords: IOR 2008, Markov Chain, Quasi-monte Carlo, Randomized Quasi-monte Carlo