Some Recent Advances in Stochastic Simulation

This review article introduces two recent advances in stochastic simulation: the construction of efficient algorithms for estimating rare events and the generation of samples from a stationary distribution that has no closed form. Estimating a very small quantity requires extreme accuracy to form a useful confidence interval. This makes the slowly convergent rare-event simulation a challenge task in both efficiency and accuracy. In this report, we introduce the examples of rare events of interest and the difficulties in estimating them. Various approaches to pursue robust and efficient estimators along the development are discussed and evaluated. Numerical experiments on estimating ruin probability are provided to show the quality of these approaches. In steady-state simulation, how to generate samples from a stationary stochastic process has long been the key subject. The common practice is to discard the data gathered during the initial transient period. However, how long the warm-up period must be raises another problem that has no satisfactory answer. Fortunately, by the development in the past two decades, exact simulation has become possible for certain stochastic models. In this report, we will introduce two important methods and related applications.