## Abstract

We study the problem of multivariate integration over R^{d} with integrands of the form f (x) ρ_{d} (x) where ρ_{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 H_{d} of functions whose mixed first derivatives, when multiplied by a weight function ψ_{d}, are bounded in the L_{2}-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.

Original language | English |
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Pages (from-to) | 630-651 |

Number of pages | 22 |

Journal | Journal of Complexity |

Volume | 22 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2006 |

### Bibliographical note

Funding Information:The authors thank Ian Sloan for many helpful comments and suggestions to the paper. The support of the Australian Research Council under its Centres of Excellence Program is gratefully acknowledged. The second author is supported by the National Sciences Foundation under Grant CCR-0511994.

## Keywords

- Quasi-Monte Carlo methods
- Randomly shifted lattice rules
- Unbounded integrands
- Worst case error

## ASJC Scopus subject areas

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- General Mathematics
- Control and Optimization
- Applied Mathematics