Control Variates for Quasi-Monte Carlo

Quasi-Monte Carlo (QMC) methods have begun to displace ordinary Monte Carlo (MC) methods in many practical problems. It is natural and obvious to combine QMC methods with traditional variance reduction techniques used in MC sampling, such as control variates. There can, however, be some surprises. The optimal control variate coefficient for QMC methods is not in general the same as for MC. Using the MC formula for the control variate coefficient can worsen the performance of QMC methods. A good control variate in QMC is not necessarily one that correlates with the target integrand. Instead, certain high frequency parts or derivatives of the control variate should correlate with the corresponding quantities of the target. We present strategies for applying control variate coefficients with QMC and illustrate the method on a 16-dimensional integral from computational finance. We also include a survey of QMC aimed at a statistical readership.