Quasi-Monte Carlo — Discrepancy between Theory and Practice

In this paper, we survey the current status of theory and practice of quasi-Monte Carlo methods. First, we discuss one of the most important research directions for accelerating Monte Carlo simulations. It is described as MC → QMC → RQMC → Derandomized RQMC, where RQMC means randomized QMC. We give some interesting open questions concerning the gap between theory and practice of derandomized RQMC. Secondly, we overview the dramatic success of quasi-Monte Carlo methods for very high dimensional numerical integration problems in finance. In the last five years, the question of how to explain this success has been extensively investigated, and two classes of problems have been identified for which QMC (or RQMC) is much more efficient than MC. One is a class of problems with small “effective” dimensions; the other is a class of isotropic problems. Some interesting results and issues on these problems are discussed.

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