On tractability of weighted integration over bounded and unbounded regions in Reals

We prove that for the space of functions with mixed rst derivatives bounded in L 1 norm, the weighted integration problem over bounded or unbounded regions is equivalent to the corresponding classical integration problem over the unit cube. This correspondence yields tractability of the general weighted integration problem. 1. Introduction In recent years there has been great interest in the tractability of multiple integration in high dimensions, much of it stimulated by the apparent success of quasi-Monte Carlo methods applied to integrals from mathematical nance over hundreds or even thousands of dimensions see, e.g., 8, 9] and 13] for more references. Most analysis has been carried out for the problem of integration over the s-dimensional unit cube in a reproducing kernel Hilbert space setting of functions whose mixed rst derivatives are square integrable, see, e.g., 7, 12]. However, as pointed out in 11], there is a fundamental diiculty in applying the Hilbert space results to the integrals from mathematical nance. These integrals are typically with respect to probability densities over unbounded regions. The diiculty (discussed in more detail below) is that after mapping to the unit cube most problems of this kind yield integrands that do not belong to the Hilbert space: the derivatives are integrable, but not square integrable. In the present paper we study the tractability of the weighted integration problem, over both bounded and unbounded regions, in the Banach space of functions whose mixed rst derivatives are in L 1 : We shall see that only in L 1 (in contrast to L p for p > 1) is there a natural correspondence between the weighted integration problems over a general region and the unweighted integration problem over the unit cube. We emphasise that in this paper we do not weight the various coordinate directions in the manner of 12]. It is known from the work of 2] that for the L 1 case the unweighted integration problem over the unit cube is tractable whereas this is believed to be not the case for p > 1; and is known 7] not to be so for p = 2: In this paper we establish the tractability of the weighted integration problem over general regions for the L 1 case by exploiting the above-mentioned connection between the weighted integration