Computational methods for complex stochastic systems: a review of some alternatives to MCMC
Statistics and Computing
Comparison of Point Sets and Sequences for Quasi-Monte Carlo and for Random Number Generation
SETA '08 Proceedings of the 5th international conference on Sequences and Their Applications
Approximate zero-variance simulation
Proceedings of the 40th Conference on Winter Simulation
A practical view of randomized quasi-Monte Carlo: invited presentation, extended abstract
Proceedings of the Fourth International ICST Conference on Performance Evaluation Methodologies and Tools
Coupling from the past with randomized quasi-Monte Carlo
Mathematics and Computers in Simulation
Quasi-Monte Carlo methods for Markov chains with continuous multi-dimensional state space
Mathematics and Computers in Simulation
Monte Carlo method for numerical integration based on Sobol's sequences
NMA'10 Proceedings of the 7th international conference on Numerical methods and applications
Simulation of coalescence with stratified sampling
Proceedings of the Winter Simulation Conference
Computers & Mathematics with Applications
American option pricing with randomized quasi-Monte Carlo simulations
Proceedings of the Winter Simulation Conference
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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.