ABSTRACT
This paper reviews recent work on numerical multiple integration over the d-dimensional unit cube with very large values of d. The work is motivated by the apparent success of recent financial derivative calculations which claim to evaluate such integrals with a small error. First, in a periodic setting that is commonly used for the study of lattice rules we show that the number of integration points required to achieve a given worst-case error is at least 2 d for every choice of integration rule. Second, for a setting commonly used to analyse quasi-Monte Carlo algorithms (namely the L2 version of the Koksma-Hlawka inequality) it is shown that the number of points needed to achieve a given worst case error is again exponential in d for every sequence of quasi-Monte Carlo algorithms. But then it is shown how a modification of the latter setting can eliminate altogether the growth in the number of points with d. The modification involves the abandonment of a usual assumption, that the behaviour is essentially the same for each of the d coordinates. Instead, the derivatives with respect to successive coordinates Eire weighted so as to reflect their declining importance.
