Information-based complexity
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Weighted tensor product algorithms for linear multivariate problems
Journal of Complexity
Complexity of weighted approximation over R
Journal of Approximation Theory
On selection criteria for lattice rules and other quasi-Monte Carlo point sets
Mathematics and Computers in Simulation - IMACS sponsored Special issue on the second IMACS seminar on Monte Carlo methods
Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces
SIAM Journal on Numerical Analysis
Worst case complexity of weighted approximation and integration over Rd
Journal of Complexity
Randomized Polynomial Lattice Rules for Multivariate Integration and Simulation
SIAM Journal on Scientific Computing
The effective dimension and quasi-Monte Carlo integration
Journal of Complexity
Sufficient conditions for fast quasi-Monte Carlo convergence
Journal of Complexity
Component-By-Component Construction of Good Intermediate-Rank Lattice Rules
SIAM Journal on Numerical Analysis
Variance Reduction via Lattice Rules
Management Science
On the convergence rate of the component-by-component construction of good lattice rules
Journal of Complexity
Journal of Complexity
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Randomly shifted lattice rules on the unit cube for unbounded integrands in high dimensions
Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
Shifted lattice rules based on a general weighted discrepancy for integrals over Euclidean space
Journal of Computational and Applied Mathematics
Journal of Complexity
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We 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.