Equivalence between Sobolev spaces of first-order dominating mixed smoothness and unanchored ANOVA spaces on ℝ^{𝕕}

Abstract

We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space onRd\mathbb {R}^d, ford≥1d \geq 1. Both spaces are Hilbert spaces involvingweight functions, which determine the behaviour as different variables tend to±∞\pm \infty, andweight parameters, which represent the influence of different subsets of variables. The unanchored ANOVA space onRd\mathbb {R}^dwas initially introduced by Nichols and Kuo in 2014 to analyse the error ofquasi-Monte Carlo(QMC) approximations for integrals on unbounded domains; whereas the classical Sobolev space of dominating mixed smoothness was used as the setting in a series of papers by Griebel, Kuo and Sloan on the smoothing effect of integration, in an effort to develop a rigorous theory on why QMC methods work so well for certain non-smooth integrands withkinksorjumpscoming from option pricing problems. In this same setting, Griewank, Kuo, Leövey and Sloan in 2018 subsequently extended these ideas by developing a practicalsmoothing by preintegrationtechnique to approximate integrals of such functions with kinks or jumps.We first prove the equivalence in one dimension (itself a non-trivial task), before following a similar, but more complicated, strategy to prove the equivalence for general dimensions. As a consequence of this equivalence, we analyse applying QMC combined with a preintegration step to approximate the fair price of an Asian option, and prove that the error of such an approximation usingNNpoints converges at a rate close to1/N1/N.