Summary
We study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain with 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. The method can be used in particular to get a lowvariance 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.
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References
F. J. Hickernell. Obtaining o(n−2+ε) convergence for lattice quadrature rules. In K.-T. Fang, F. J. Hickernell, and H. Niederreiter, editors, Monte Carlo and Quasi-Monte Carlo Methods 2000, pages 274–289, Berlin, 2002. Springer-Verlag.
C. Lécot and S. Ogawa. Quasirandom walk methods. In K.-T. Fang, F. J. Hickernell, and H. Niederreiter, editors, Monte Carlo and Quasi-Monte Carlo Methods 2000, pages 63–85, Berlin, 2002. Springer-Verlag.
C. Lécot and B. Tuffin. Quasi-Monte Carlo methods for estimating transient measures of discrete time Markov chains. In H. Niederreiter, editor, Monte Carlo and Quasi-Monte Carlo Methods 2002, pages 329–343, Berlin, 2004. Springer-Verlag.
P. L’Ecuyer. SSJ: A Java Library for Stochastic Simulation, 2004. Software user’s guide, Available at http://www.iro.umontreal.ca/~lecuyer.
P. L’Ecuyer and C. Lemieux. Variance reduction via lattice rules. Management Science, 46(9):1214–1235, 2000.
P. L’Ecuyer and C. Lemieux. Recent advances in randomized quasi-Monte Carlo methods. In M. Dror, P. L’Ecuyer, and F. Szidarovszky, editors, Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pages 419–474. Kluwer Academic Publishers, Boston, 2002.
C. Lemieux and P. L’Ecuyer. A comparison of Monte Carlo, lattice rules and other low-discrepancy point sets. In H. Niederreiter and J. Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods 1998, pages 326–340, Berlin, 2000. Springer-Verlag.
A. B. Owen. Latin supercube sampling for very high-dimensional simulations. ACM Transactions on Modeling and Computer Simulation, 8(1):71–102, 1998.
A. B. Owen. Variance with alternative scramblings of digital nets. ACM Transactions on Modeling and Computer Simulation, 13(4):363–378, 2003.
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L’Ecuyer, P., Lécot, C., Tuffin, B. (2006). Randomized Quasi-Monte Carlo Simulation of Markov Chains with an Ordered State Space. In: Niederreiter, H., Talay, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31186-6_19
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DOI: https://doi.org/10.1007/3-540-31186-6_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25541-3
Online ISBN: 978-3-540-31186-7
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