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.
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.