Randomly shifted lattice rules for unbounded integrands

AbstractWe study the problem of multivariate integration over Rd with integrands of the form f(x)ρd(x) where ρd 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 ψd, are bounded in the L2-norm. We prove that good randomly shifted lattice rules can be constructed component by component to achieve a worst case error of order O(n-1/2), where the implied constant can be independent of d. We experiment with the Asian option problem using the rules constructed in several variants of the new function space. Our results are as good as those obtained in the anchored Sobolev spaces and they are significantly better than those obtained by the Monte Carlo method