The sparse structure of high-dimensional integrands

Abstract

We consider (a discretization of) a functional of white noise over a finite time interval. We explore the possible interest of representing the white noise in the orthonormal bases of orthogonal polynomials or wavelets for the numerical evaluation of the expected value of this functional. Using the Wiener–Itô decomposition of the functional, the sparsity is studied of the representation of the functional in these bases. An approximation scheme is proposed that uses existing low-dimensional quasi-Monte Carlo rules and takes profit of the sparse structure of the quadratic part of the functional.