Wegelerstraße 6 • 53115 Bonn • Germany phone +49 228 73-3427 • fax +49 228 73-7527 www.ins.uni-bonn.de Michael Griebel, Frances Y. Kuo, and Ian H. Sloan The Smoothing Effect of integration in Rd and the ANOVA Decomposition INS Preprint No. 1007 December 2010 The smoothing effect of integration in Rd and the ANOVA decomposition Michael Griebel,∗ Frances Y. Kuo,y and Ian H. Sloanz May 2011 Abstract This paper studies the ANOVA decomposition of a d-variate function f defined on the whole of Rd, 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. 2010 Mathematics Subject Classification: Primary 41A63, 41A99. Secondary 65D30. Keywords: ANOVA decomposition, smoothing, option pricing. 1 Introduction In this paper we study the ANOVA decomposition of d-variate real-valued functions f defined on the whole of Rd, where f fails to be smooth because it is the maximum of a smooth function and zero. That is, we consider d f(x) = ϕ(x)+ := max(ϕ(x); 0); x 2 R ; (1) with ϕ a smooth function on Rd. The conclusions will apply equally to the absolute value of ϕ, since jϕ(x)j = ϕ(x)+ + (−ϕ(x))+: Our study is motivated by option pricing problems, which take the form of (1) because a financial option is considered to be worthless once its value drops below a specified `strike price'. ∗Institut f¨ur Numerische Simulation, Wegelerstreet 6, 53115, Bonn, Germany. Email: [email protected] ySchool of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia. Email: [email protected] (corresponding author) zSchool of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia. Email: [email protected] 1 In a previous paper [8] we considered the smoothness of the terms of the ANOVA decomposition when a d-variate function such as (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). 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. In the present paper we avoid the mapping to the unit cube, and instead treat the problem as one posed on the whole of Rd. In this case the results turn out to be more sur- prising, in that the effect of integrating out a single variable can be unlimited smoothness with respect to the other variables, in contrast to an increase in smoothness of just one degree in the case of the unit cube. These results are expected to lay the foundation for a future rigorous error analysis of direct numerical methods for option pricing integrals over Rd, methods that do not involve mapping Rd to the unit cube. The structure of the paper is as follows. In Section 2 we establish the mathematical background, including the definition of the ANOVA decomposition, and define the nota- tion. In Section 3 we demonstrate the smoothing effect produced by integrating out a single variable. In Section 4 we apply the results to the problem of pricing Asian options, with the striking result that, in the case of both the standard and Brownian bridge con- structions, every term of the ANOVA decomposition except for the very highest one has unlimited smoothness. Numerical examples in Section 5 complete the paper. 2 Background Let ρ be a continuous andR strictly positive univariate probability density function, i.e., 2 R 1 ρ(t) > 0 for all t and −∞ ρ(t) dt = 1. From this we construct a d-variate probability density Yd d ρd(x) := ρ(xj) for x = (x1; : : : ; xd) 2 R : j=1 2 1 L Rd L Rd For p [1; ], we consider the weighted p space defined over , denoted by p,ρd ( ), with the weighted norm ((R ) j jp 1=p 2 1 Rd f(x) ρd(x) dx if p [1; ); kfkL = (2) p,ρd j j 1 ess supx2Rd f(x) if p = : k k ≤ k k ≤ 0 It can be verified using H¨older'sinequality that f Lp,ρ f L 0 for p p , and hence d p ,ρd d d d 0 L 0 R ⊆ L R ⊆ L R ≤ ≤ ≤ 1 p ,ρd ( ) p,ρd ( ) 1,ρd ( ) for 1 p p : (3) 2 d If a function f defined on R is integrable with respect to ρd, i.e., if f 2 L1,ρ , we write Z d Idf := f(x) ρd(x) dx: Rd Then Z jI fj ≤ jf(x)j ρ (x) dx = kfkL : d d 1,ρd Rd Throughout this paper we assume that the dimension d is fixed, and we write D := f1; 2; : : : ; dg: 2.1 Univariate integration and the ANOVA decomposition 2 2 L Rd For j D and f 1,ρd ( ), let Pj be the projection defined by Z 1 d (Pjf)(x) = f(x1; : : : ; xj−1; tj; xj+1; : : : ; xd) ρ(tj) dtj for x = (x1; : : : ; xd) 2 R : −∞ Thus Pjf is the function obtained by integrating out the jth component of x with respect to the weight function ρ, and so is a function that is constant with respect to xj. For convenience we often say that Pjf does not depend on this component xj, and we write interchangeably (Pjf)(x) = (Pjf)(xDnfjg); where xDnfjg denotes the d − 1 components of x apart from xj, and we express the corresponding (d − 1)-dimensional Euclidean space by RDnfjg. By Fubini's theorem [5, L RDnfjg Section 5.4], Pjf exists for almost all xDnfjg and belongs to 1,ρDnfjg ( ). For u ⊆ D we write Y Pu = Pj: j2u Here the ordering within the product is not important because, by Fubini's theorem, PjPk = PkPj for all j; k 2 D. Thus Puf is the function obtained by integrating out all 2 the components of x with indices in u. Note that Pu = Pu and PD = Id. The ANOVA decomposition of f (see, e.g., [6, 12]) is X f = fu; u⊆D with fu depending only on the variables xj with indices j 2 u, and with fu satisfying Pjfu = 0 for all j 2 u. The functions fu satisfy the recurrence relation X f; = Idf and fu = PDnuf − fv: v(u Often this recurrence relation is used as the defining property of the ANOVA terms fu. It is known, for example from the recent paper [11], that the ANOVA terms fu are given explicitly by X X ju|−|vj ju|−|vj fu = (−1) PDnvf = PDnuf + (−1) Punv(PDnuf): (4) v⊆u v(u 3 In the latter form it becomes plausible that the smoothness of fu is determined by PDnuf, since we do not expect the further integrations Punv in the terms of the second sum to reduce the smoothness of PDnuf; this expectation is proved in Theorem 2 below. 2.2 Sobolev spaces and weak derivatives For j 2 D, let Dj denote the partial derivative operator @f (Djf)(x) = (x): @xj Throughout this paper, the term multi-index refers to a vector α = (α1; : : : ; αd) whose components are nonnegative integers, and we use the notation jαj = α1 + ··· + αd to denote the sum of its components. For any multi-index α = (α1; : : : ; αd), we define ( ) Yd Yd @ αj @jαj Dα = Dαj = = Q ; (5) j @x d αj j=1 j=1 j j=1 @xj and we say that the derivative Dαf is of order jαj. Let C(Rd) = C0(Rd) denote the linear space of continuous functions defined on Rd. For a nonnegative integer r ≥ 0, we define Cr(Rd) to be the space of functions whose classical d derivatives of order ≤ r are all continuous at every point in R , withP no limitation on d 2 their behaviour at infinity. For example, the function f(x) = exp( j=1 xj ) belongs to r d 1 d r d C (R ) for all values of r. For convenience we write C (R ) = \r≥0C (R ). In addition to classical derivatives, we shall consider also weak derivatives in this paper. By definition, the weak derivative Dαf is a measurable function on Rd which satisfies Z Z α − jαj α 2 C1 Rd (D f)(x) v(x) dx = ( 1) f(x)(D v)(x) dx for all v 0 ( ); (6) Rd Rd C1 Rd where 0 ( ) denotes the space of infinitely differentiable functions with compact sup- port in Rd, and where the derivatives on the right-hand side of (6) are classical partial derivatives.