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

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 on $\mathbb {R}^d$, for $d \geq 1$. Both spaces are Hilbert spaces involving weight functions, which determine the behaviour as different variables tend to $\pm \infty$, and weight parameters, which represent the influence of different subsets of variables. The unanchored ANOVA space on $\mathbb {R}^d$ was initially introduced by Nichols and Kuo in 2014 to analyse the error of quasi-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 with kinks or jumps coming from option pricing problems. In this same setting, Griewank, Kuo, Leövey and Sloan in 2018 subsequently extended these ideas by developing a practical smoothing by preintegration technique 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 using $N$ points converges at a rate close to $1/N$.