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The smoothing effect of the ANOVA decomposition by Michael Griebel; Frances Y. Kuo; Ian H. Sloan is a Mathematics article available to read on EtoBox.

What is The smoothing effect of the ANOVA decomposition about?

We show that the lower-order terms in the ANOVA decomposition of a function f (x) := max(φ(x), 0) for x ∈ [0, 1] d , with φ a smooth function, may be smoother than f itself. Specifically, f in general belongs only to W 1 d,∞ , i.e., f has one essentially bounded derivative with respect to any component of x, whereas, for each u ⊆ {1, . . . , d}, the ANOVA term f u (which depends only on the variables x j with j ∈ u) belongs to W 1+τ d,∞ , where τ is the number of indices k ∈ {1, . . . , d} \ u for which ∂φ/∂x k is never zero. As an application, we consider the integrand arising from pricing an arithmetic Asian option on a single stock with d time intervals. After transformation of the integral to the unit cube and also employing a boundary truncation strategy, we show that for both the standard and the Brownian bridge constructions of the paths, the ANOVA terms that depend on (d+1)/2 or fewer variables all have essentially bounded mixed first derivatives; similar but slightly weaker results hold for the principal components construction. This may explain why quasi-Monte Carlo and sparse grid approximations of option pricing integrals often exhibit nearly first order convergence, in

Who reads The smoothing effect of the ANOVA decomposition?

Author

Michael Griebel; Frances Y. Kuo; Ian H. Sloan

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0885-064X)

Published

2010

Language

EN

Field

Mathematics (Physical Sciences)