The effective dimension and quasi-Monte Carlo integration

Abstract

Quasi-Monte Carlo (QMC) methods are successfully used for high-dimensional integrals arising in many applications. To understand this success, the notion of effective dimension has been introduced. In this paper, we analyse certain function classes commonly used in QMC methods for empirical and theoretical investigations and show that the problem of determining their effective dimension is analytically tractable. For arbitrary square integrable functions, we propose a numerical algorithm to compute their truncation dimension. We also consider some realistic problems from finance: the pricing of options. We study the special structure of the corresponding integrands by determining their effective dimension and show how large the effective dimension can be reduced and how much the accuracy of QMC estimates can be improved by using the Brownian bridge and the principal component analysis techniques. A critical discussion of the influence of these techniques on the QMC error is presented. The connection between the effective dimension and the performance of QMC methods is demonstrated by examples.