Non-Uniform Low-Discrepancy Sequence Generation and Integration of Singular Integrands

Jürgen Hartinger () and Reinhold Kainhofer ()
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Jürgen Hartinger: Graz University of Technology, Department of Mathematics
Reinhold Kainhofer: Vienna University of Technology, Department of Math. Methods in Economics

A chapter in Monte Carlo and Quasi-Monte Carlo Methods 2004, 2006, pp 163-179 from Springer

Abstract: Summary In this article, we will first highlight a method proposed by Hlawka and Mück to generate low-discrepancy sequences with an arbitrary distribution H and discuss its shortcomings. As an alternative, we propose an interpolated inversion method that is also shown to generate H-distributed low-discrepancy sequences, in an effort of order O(N log N). Finally, we will address the issue of integrating functions with a singularity on the boundaries. Sobol’ and Owen proved convergence theorems and orders for the uniform distribution, which we will extend to general distributions. Convergence orders will be proved under certain origin- or corner-avoidance conditions, as well as growth conditions on the integrand and the density. Our results prove that also non-uniform quasi-Monte Carlo methods can be well applied to integrands with a polynomial singularity at the integration boundaries.

Keywords: Singular Integration; Support Point; Error Order; Asian Option; Hyperbolic Region (search for similar items in EconPapers)
Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-31186-7_11

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DOI: 10.1007/3-540-31186-6_11

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