Variance reduction via basis expansion in Monte Carlo integration

<!-- *** Custom HTML *** --> Monte Carlo methods are widely used in numerical integration, and variance reduction plays a key role in Monte Carlo integration. This paper investigates variance reduction for Monte Carlo integration in both finite dimensional Euclidean space and infinite dimensional Wiener space. The proposed variance reduction approaches are to use basis functions to construct control variates for finite dimensional integrals and utilize Itô-Wiener chaos expansion to design antithetic variates and control variates for Wiener integrals. We establish the variances of the proposed Monte Carlo integration estimators and show that the proposed methods can achieve dramatic variance reduction in comparison with the basic Monte Carlo estimators. Examples are used to illustrate the performance of the proposed estimators.