On Accelerating Monte Carlo Integration Using Orthogonal Projections

Monte Carlo simulation is an indispensable tool in calculating high-dimensional integrals. Although Monte Carlo integration is notoriously known for its slow convergence, it could be improved by various variance reduction techniques. This paper applies orthogonal projections to study the amount of variance reduction, and also proposes a novel projection estimator that is associated with a group of symmetries of the probability measure. For a given space of functions, the average variance reduction can be derived. For a specific function, its variance reduction is also analyzed. The well-known antithetic estimator is a special case of the projection estimator, and new results of its variance reduction and efficiency are provided. Various illustrations including pricing financial Asian options are provided to confirm our claims.