Uniformly Efficient Rare Event Simulation

Specialeforsvar ved Anders Rechter

Titel: Uniformly Efficient Rare Event Simulation 

Abstract: Rare event simulation is typically only efficient in a single event. An example of this is the estimation of the tail probability P(S_n\ge n*a) for some a, where S_n=\sum_{i=1}^n X_i is a random walk and X_1,...,X_n are iid and light tailed. Importance sampling with sampling from a single exponentially twisted measure can be used to estimate P(S_n\ge n*a). This will only be asymptotically efficient for a single a so it is unsuitable for estimation of P(S_n \ge n*a) for all a in A, if A contains more than one point. Importance sampling with sampling from a mixture of k exponentially twisted measures can estimate P(S_n\ge n\cdot a) asymptotically efficient when k goes to infinity as n goes to infinity for all a in A, where A is an interval. We will prove that the method achieves asymptotic efficiency, given that the measures are chosen correctly, as well as give a method of calculating the measures. These ideas and proofs were first proposed in an article by P. Glasserman and S. Juneja \cite{glasserman}. We will discuss the appli-cation of these ideas to the problem of quantile estimation and compare with P. Glynn's article on quantile estimation \cite{glynn}. We will perform a simulation experiment and show efficiency empirically. We will adapt the ideas from the random walk case to the problem of estimating the probability of early ruin over a large time interval. We will show empirically that efficiency is achieved.

Vejleder: Jeffrey Collamore
Censor:  Christian Tarp, Codan Forsikring