Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Quasi-Monte Carlo integration over $\mathbb{R}^d$

Authors: Peter Mathé and Gang Wei
Journal: Math. Comp. 73 (2004), 827-841
MSC (2000): Primary 65C05; Secondary 68Q25
Published electronically: August 7, 2003
MathSciNet review: 2031408
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we show that a wide class of integrals over $\mathbb R^d$ with a probability weight function can be evaluated using a quasi-Monte Carlo algorithm based on a proper decomposition of the domain $\mathbb R^d$ and arranging low discrepancy points over a series of hierarchical hypercubes. For certain classes of power/exponential decaying weights the algorithm is of optimal order.

References [Enhancements On Off] (What's this?)

  • 1. F. Delbaen.
    Coherent risk measures on general probability spaces. http://www.math. $\tilde{\hspace{1ex}}$delbaen/, 1998.
  • 2. R. E. Edwards.
    Functional Analysis. Theory and Applications.
    Holt, Rinehart & Winston, New York, Chicago, 1965.
  • 3. L.K. Hua and Y. Wang
    Applications of Number Theory to Numerical Analysis.
    Springer-Verlag, Berlin, New York, 1981. MR 83g:10034
  • 4. Norbert Hofmann and Peter Mathé.
    On quasi-Monte Carlo simulation of stochastic differential equations.
    Math. Comp., 66(218):573-589, 1997. MR 97f:65005
  • 5. Stefan Jaschke and Uwe Küchler.
    Coherent risk measures and good-deal bounds.
    Finance Stoch., 5(2):181-200, 2001. MR 2003a:91066
  • 6. H. Niederreiter.
    Random Number Generation and Quasi-Monte Carlo Methods, volume 63 of CBMS-NSF Region. Conf. Series in Appl. Math., SIAM, Philadelphia, 1992. MR 93h:65008
  • 7. A. Papageorgiou.
    Fast convergence of quasi-Monte Carlo for a class of isotropic integrals.
    Math. Comp., 70(233):297-306, 2001. MR 2001f:65008
  • 8. -. $\tilde{\hspace{1ex}}$ap/html/finder.html.
  • 9. L. Plaskota, K. Ritter and G. W. Wasilkowski.
    Average case complexity of weighted integration and approximation over $\mathbb R^d$ with isotropic weight.
    In Monte Carlo and quasi-Monte Carlo Methods 2000 (K.-T. Fang, F. J. Hickernell, and H. Niederreiter, eds.), pp. 446-459, Springer-Verlag, Berlin, 2002.
  • 10. I. M. Sobol$'$.
    An exact bound for the error of multivariate integration formulas for functions of classes $\tilde{W}\sb{1}$ and $\tilde{H}\sb{1}$.
    Z. Vycisl. Mat. i Mat. Fiz., 1:208-216, 1961. MR 24:B2546
  • 11. William P. Ziemer.
    Weakly differentiable functions.
    Springer-Verlag, New York, 1989. MR 91e:46046

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65C05, 68Q25

Retrieve articles in all journals with MSC (2000): 65C05, 68Q25

Additional Information

Peter Mathé
Affiliation: Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, D-10117 Berlin, Germany

Gang Wei
Affiliation: Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong

Keywords: Quasi--Monte Carlo integration, elliptically contoured distributions
Received by editor(s): June 11, 2002
Received by editor(s) in revised form: October 10, 2002
Published electronically: August 7, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society