Randomly shifted lattice rules for unbounded integrands

We study the problem of multivariate integration over Rd with integrands of the form f(x)d(x) where d is a probability density function. Practical problems of this form occur commonly in statistics and mathematical finance. The necessary step before applying any quasi-Monte Carlo method is to transform the integral into the unit cube [0, 1]d. However, such transformations often result in integrands which are unbounded near the boundary of the cube, and thus most of the existing theory on quasi-Monte Carlo methods cannot be applied. In this paper we assume that f belongs to some weighted tensor product reproducing kernel Hilbert space Hd of functions whose mixed first derivatives, when multiplied by a weight function d, are bounded in the L2-norm. By exploiting the isometry between Hd and the corresponding space of transformed integrands defined over (0, 1)d, we proved that good randomly-shifted lattice rules can be constructed component-by-component to achieve a worst case error of or...

• Type: Journal
• Year: 2006
• Where: JC
• Authors: Frances Y. Kuo, Grzegorz W. Wasilkowski, Benjamin J. Waterhouse
• Keywords: Hilbert Space Hd, JC 2006, Probability Density Function, Quasi-monte Carlo Methods