Adaptive Monte Carlo Variance Reduction with Two-time-scale Stochastic Approximation

Combined control variates and importance sampling variance reduction and its two-fold optimality are investigated. Two-time-scale stochastic approximation algorithm is applied in parameter search for the combination and almost sure convergence of the algorithm to the unique optimum is proved. The parameter search procedure is further incorporated into adaptive Monte Carlo simulation, and its law of large numbers and central limit theorem are proved to hold. An numerical example is provided to illustrate the effectiveness of the method.