The smoothing effect of the ANOVA decomposition

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 ϕϕ a smooth function, may be smoother than ff itself. Specifically, ff in general belongs only to Wd,∞1, i.e., ff has one essentially bounded derivative with respect to any component of x, whereas, for each u⊆{1,…,d}, the ANOVA term fu (which depends only on the variables xjxj with j∈u) belongs to Wd,∞1+τ, where ττ is the number of indices k∈{1,…,d}∖u for which ∂ϕ/∂xk∂ϕ/∂xk is never zero.As an application, we consider the integrand arising from pricing an arithmetic Asian option on a single stock with dd 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(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 spite of lacking the smoothness required by the conventional theories.