Randomly shifted lattice rules on the unit cube for unbounded integrands in high dimensions

  • Authors:

  • Ben J. Waterhouse;Frances Y. Kuo;Ian H. Sloan

  • Affiliations:

  • School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia;School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia;School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia

  • Venue:
  • Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
  • Year:
  • 2006

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Abstract

We study the problem of multivariate integration on the unit cube for unbounded integrands. Our study is motivated by problems in statistics and mathematical finance, where unbounded integrands can arise as a result of using the cumulative inverse normal transformation to map the integral from the unbounded domain R^d to the unit cube [0,1]^d. We define a new space of functions which possesses the boundary behavior of those unbounded integrands arising from statistical and financial applications, however, we assume that the functions are analytic, which is not usually the case for functions from finance problems. Our new function space is a weighted tensor-product reproducing-kernel Hilbert space. We carry out a worst-case analysis in this space and show 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 in the big O notation is independent of d if the sum of the weights is finite. Numerical experiments indicate that our lattice rules are reasonably robust and perform well in pricing Asian options.