Approximate Zero-Variance Simulation

Abstract: 

Monte Carlo methods provide estimators whose standard error usually decreases at a slow rate dictated by the central limit theorem. In theory, there exist simulation estimators with zero variance, i.e., that always provide the exact value. The catch is that these estimators are usually much too difficult (or virtually impossible) to implement.

However, there are situations, especially in the context of rare-event simulation, where the zero-variance simulation can be approximated well enough to provide huge efficiency gains. Adaptive versions can even yield a faster convergence rate, including exponential convergence in some cases. We give an overview of these methods, discuss their practicality, and provide concrete numerical examples.