The smoothing effect of integration in $\mathbb{R}d$ and the ANOVA decomposition

This paper studies the ANOVA decomposition of a $ d$-variate function $ f$ defined on the whole of $ \mathbb{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 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.