Quasi–Monte Carlo integration over ℝ^{𝕕}

Mathé, Peter; Wei, Gang

doi:10.1090/S0025-5718-03-01569-2

Quasi–Monte Carlo integration over $\mathbb {R}^d$
HTML articles powered by AMS MathViewer

by Peter Mathé and Gang Wei PDF

Math. Comp. 73 (2004), 827-841 Request permission

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

F. Delbaen. Coherent risk measures on general probability spaces. http://www.math. ethz.ch/$\tilde {\hspace {1ex}}$delbaen/, 1998.
R. E. Edwards. Functional Analysis. Theory and Applications. Holt, Rinehart & Winston, New York, Chicago, 1965.
Loo Keng Hua and Yuan Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin-New York; Kexue Chubanshe (Science Press), Beijing, 1981. Translated from the Chinese. MR 617192
Norbert Hofmann and Peter Mathé, On quasi-Monte Carlo simulation of stochastic differential equations, Math. Comp. 66 (1997), no. 218, 573–589. MR 1397444, DOI 10.1090/S0025-5718-97-00820-X
Stefan Jaschke and Uwe Küchler, Coherent risk measures and good-deal bounds, Finance Stoch. 5 (2001), no. 2, 181–200. MR 1841716, DOI 10.1007/PL00013530
Harald Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1172997, DOI 10.1137/1.9781611970081
A. Papageorgiou, Fast convergence of quasi-Monte Carlo for a class of isotropic integrals, Math. Comp. 70 (2001), no. 233, 297–306. MR 1709157, DOI 10.1090/S0025-5718-00-01231-X
—. http://www.cs.columbia.edu/$\tilde {\hspace {1ex}}$ap/html/finder.html.
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.
I. M. Sobol′, An exact bound for the error of multivariate integration formulas for functions of classes $\~W_{1}$ and $\~H_{1}$, Ž. Vyčisl. Mat i Mat. Fiz. 1 (1961), 208–216 (Russian). MR 136513
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3