Adaptive Control Variates in Monte Carlo Simulation

Monte Carlo simulation is widely used in many fields. Unfortunately, it usually requires a large amount of computer time to obtain even moderate precision so it is necessary to apply efficiency improvement techniques. Adaptive Monte Carlo methods are specialized Monte Carlo simulation techniques where the methods are adaptively tuned as the simulation progresses. The primary focus of such techniques has been in adaptively tuning importance sampling distributions to reduce the variance of an estimator. We instead focus on adaptive methods based on control variate schemes. In this dissertation we introduce two adaptive control variate methods where a family of parameterized control variates is available, and develop their asymptotic properties. The first method is based on a stochastic approximation scheme for identifying the optimal choice of control variate. It is easily implemented, but its performance is sensitive to certain tuning parameters, the selection of which is nontrivial. The second method uses a sample average approximation approach. It has the advantage that it does not require any tuning parameters, but it can be computationally expensive and requires the availability of nonlinear optimization software. We include implementations of the methods and numerical results for two applications. These results suggests that the adaptive methods outperform the na¨ıve approach as long as the parameterization of the control variate is carefully chosen.