MATHEMATICS OF COMPUTATION Volume 82, Number 281, January 2013, Pages 383–400 S 0025-5718(2012)02578-6 Article electronically published on July 20, 2012 THE SMOOTHING EFFECT OF INTEGRATION IN R d AND THE ANOVA DECOMPOSITION MICHAEL GRIEBEL, FRANCES Y. KUO, AND IAN H. SLOAN Abstract. This paper studies the ANOVA decomposition of a d-variate func- tion f defined on the whole of 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 condi- tions 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. 1. Introduction In this paper we study the ANOVA decomposition of d-variate real-valued func- tions f defined on the whole of R d , where f fails to be smooth because it is the maximum of a smooth function and zero. That is, we consider (1.1) f (x)= φ(x) + := max(φ(x), 0), x ∈ R d , with φ a smooth function on R d . The conclusions will apply equally to the absolute value of φ, since |φ(x)| = φ(x) + +(−φ(x)) + . Our study is motivated by option pricing problems, which take the form of (1.1) because a financial option is considered to be worthless once its value drops below a specified ‘strike price’. In a previous paper [8] we considered the smoothness of the terms of the ANOVA decomposition when a d-variate function such as (1.1) is mapped to the unit cube in a suitable way. There we found, under suitable conditions, that the low-order terms of the ANOVA decomposition can be reasonably smooth, even though f itself has a ‘kink’ arising from the max function in (1.1). Essentially, this occurs because the process of integrating out the ‘other’ variables has a smoothing effect. The smoothness matters if quasi-Monte Carlo [13, 14] or sparse grid [4] methods are used to estimate the expected values of financial options expressed as high dimensional integrals, because the convergence theory for both of these methods assumes that the integrands have (at least) square integrable mixed first derivatives [7, 10], a property that is manifestly not true for the ‘kink’ function. But a rigorous error analysis becomes thinkable if, on the one hand, the higher order ANOVA terms are small (as is often speculated to be the case—this is the notion of ‘low superposition dimension’ introduced by [6]), and on the other hand, if the low-order ANOVA terms all have the required smoothness property. Received by the editor December 14, 2010 and, in revised form, May 31, 2011. 2010 Mathematics Subject Classification. Primary 41A63, 41A99; Secondary 65D30. Key words and phrases. ANOVA decomposition, smoothing, option pricing. c 2012 American Mathematical Society 383 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use