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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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


Authors: Michael Griebel, Frances Y. Kuo and Ian H. Sloan
Journal: Math. Comp. 82 (2013), 383-400
MSC (2010): Primary 41A63, 41A99; Secondary 65D30
Published electronically: July 20, 2012
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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.


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Additional Information

Michael Griebel
Affiliation: Institut für Numerische Simulation, Wegelerstreet 6, 53115, Bonn, Germany
Email: griebel@ins.uni-bonn.de

Frances Y. Kuo
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
Email: f.kuo@unsw.edu.au

Ian H. Sloan
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
Email: i.sloan@unsw.edu.au

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02578-6
PII: S 0025-5718(2012)02578-6
Keywords: ANOVA decomposition, smoothing, option pricing
Received by editor(s): December 14, 2010
Received by editor(s) in revised form: May 31, 2011
Published electronically: July 20, 2012
Article copyright: © Copyright 2012 American Mathematical Society