Some Recent Results in Rare Event Estimation

This article presents several state-of-the-art Monte Carlo methods for simulating and estimating rare events. A rare event occurs with a very small probability, but its occurrence is important enough to justify an accurate study. Rare event simulation calls for specific techniques to speed up standard Monte Carlo sampling, which requires unacceptably large sample sizes to observe the event a sufficient number of times. Among these variance reduction methods, the most prominent ones are Importance Sampling (IS) and Multilevel Splitting, also known as Subset Simulation. This paper offers some recent results on both aspects, motivated by theoretical issues as well as by applied problems.