Quasi-random numbers and their applications: using lattice rules for variance reduction in simulation

Quasi-Monte Carlo methods are designed to improve upon the Monte Carlo method for multidimensional numerical integration by using a more regularly distributed point set than the i.i.d. sample associated with Monte Carlo. Lattice rules are one family of quasi-Monte Carlo methods, originally proposed by Korobov in 1959. In this paper, we explain how randomized lattice rules can be used to construct efficient estimators for typical simulation problems, and we give several numerical examples. We are interested in two main aspects: Studying the variance of these estimators and finding which properties of the lattice rules should be considered when defining a selection criterion to rate and choose them. Our numerical results for three different problems illustrate how this methodology typically improves upon the usual Monte Carlo simulation method.