Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling

Abstract

In this paper we introduce a reproducing kernel Hilbert space defined on Rd+1 as the tensor product of a reproducing kernel defined on the unit sphere Sd in Rd+1 and a reproducing kernel defined on [0,∞). We extend Stolarsky’s invariance principle to this case and prove upper and lower bounds for numerical integration in the corresponding reproducing kernel Hilbert space.The idea of separating the direction from the distance from the origin can also be applied to the construction of quadrature methods. An extension of the area-preserving Lambert transform is used to generate points on Sd−1 via lifting Sobol’ points in [0,1)d to the sphere. The dth component of each Sobol’ point, suitably transformed, provides the distance information so that the resulting point set is normally distributed in Rd.Numerical tests provide evidence of the usefulness of constructing Quasi-Monte Carlo type methods for integration in such spaces. We also test this method on examples from financial applications (option pricing problems) and compare the results with traditional methods for numerical integration in Rd.