The smoothing effect of integration in ℝ^{𝕕} and the ANOVA decomposition

Griebel, Michael; Kuo, Frances; Sloan, Ian

doi:10.1090/S0025-5718-2012-02578-6

The smoothing effect of integration in $\mathbb {R}^d$ and the ANOVA decomposition
HTML articles powered by AMS MathViewer

by Michael Griebel, Frances Y. Kuo and Ian H. Sloan PDF

Math. Comp. 82 (2013), 383-400 Request permission

Corrigendum: Math. Comp. 86 (2017), 1847-1854.

Abstract:

This paper studies the ANOVA decomposition of a $d$-variate function $f$ defined on the whole of $\mathbb {R}^d$, where $f$ is the maximum of a smooth function and zero (or $f$ could be the absolute value of a smooth function). Our study is motivated by option pricing problems. We show that under suitable conditions all terms of the ANOVA decomposition, except the one of highest order, can have unlimited smoothness. In particular, this is the case for arithmetic Asian options with both the standard and Brownian bridge constructions of the Brownian motion.

References

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Kendall Atkinson and Weimin Han, Theoretical numerical analysis, Texts in Applied Mathematics, vol. 39, Springer-Verlag, New York, 2001. A functional analysis framework. MR 1817388, DOI 10.1007/978-0-387-21526-6
Hans-Joachim Bungartz and Michael Griebel, Sparse grids, Acta Numer. 13 (2004), 147–269. MR 2249147, DOI 10.1017/S0962492904000182
J. C. Burkill, The Lebesgue integral, Cambridge Tracts in Mathematics and Mathematical Physics, No. 40, Cambridge, at the University Press, 1951. MR 0045196
R. E. Caflisch, W. Morokoff, and A. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance 1, 27–46 (1997).
Michael Griebel, Sparse grids and related approximation schemes for higher dimensional problems, Foundations of computational mathematics, Santander 2005, London Math. Soc. Lecture Note Ser., vol. 331, Cambridge Univ. Press, Cambridge, 2006, pp. 106–161. MR 2277104, DOI 10.1017/CBO9780511721571.004
Michael Griebel, Frances Y. Kuo, and Ian H. Sloan, The smoothing effect of the ANOVA decomposition, J. Complexity 26 (2010), no. 5, 523–551. MR 2719646, DOI 10.1016/j.jco.2010.04.003
Steven G. Krantz and Harold R. Parks, The implicit function theorem, Birkhäuser Boston, Inc., Boston, MA, 2002. History, theory, and applications. MR 1894435, DOI 10.1007/978-1-4612-0059-8
Frances Y. Kuo and Ian H. Sloan, Lifting the curse of dimensionality, Notices Amer. Math. Soc. 52 (2005), no. 11, 1320–1329. MR 2183869
F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski, and H. Woźniakowski, On decompositions of multivariate functions, Math. Comp. 79 (2010), no. 270, 953–966. MR 2600550, DOI 10.1090/S0025-5718-09-02319-9
Ruixue Liu and Art B. Owen, Estimating mean dimensionality of analysis of variance decompositions, J. Amer. Statist. Assoc. 101 (2006), no. 474, 712–721. MR 2281247, DOI 10.1198/016214505000001410
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
I. H. Sloan and S. Joe, Lattice methods for multiple integration, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1442955