### A Randomized Quasi-Monte Carlo Simulation Method for Markov ChainsA Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

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 Author L'Ecuyer, P. ♦ Lécot, Christian ♦ Tuffin, Bruno Source Hyper Articles en Ligne (HAL) Content type Text File Format PDF Language English
 Subject Keyword MARKOV PROCESSES ♦ SIMULATION ♦ RANDOMIZED QUASI-MONTE CARLO ♦ info ♦ Computer Science [cs]/Other [cs.OH] Abstract We introduce and study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain, under the assumption that the chain has a totally ordered (discrete or continuous) state space. 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 low-discrepancy 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. We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial. The variance is reduced by factors of several thousands in some cases. We prove bounds on the convergence rate of the worst-case error and variance for special situations. In line with what is typically observed in RQMC contexts, our empirical results indicate much better convergence than what these bounds guarantee. Educational Use Research Learning Resource Type Report ♦ Article Publisher Date 2005-01-01 Publisher Institution INRIA