Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Walsh Spaces Containing Smooth Functions and Quasi-Monte Carlo Rules of Arbitrary High Order
SIAM Journal on Numerical Analysis
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We show that the lower-order terms in the ANOVA decomposition of a function f(x)@?max(@f(x),0) for x@?[0,1]^d, with @f a smooth function, may be smoother than f itself. Specifically, f in general belongs only to W"d","~^1, i.e., f has one essentially bounded derivative with respect to any component of x, whereas, for each u@?{1,...,d}, the ANOVA term f"u (which depends only on the variables x"j with j@?u) belongs to W"d","~^1^+^@t, where @t is the number of indices k@?{1,...,d}@?u for which @?@f/@?x"k is never zero. As an application, we consider the integrand arising from pricing an arithmetic Asian option on a single stock with d 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 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.