Failure Probabilities and Regular Variation for Stochastic Processes 
Henrik Hult, PhD
School of Operations Research and Industrial Engineering
Cornell University
Thursday, March 2, 2006
4:10 p.m.
E105 Engineering Building
ABSTRACT
In various applications we are interested in the probability of system failure. For instance ruin of an insurance company, large losses in a financial market or long waiting times in a computer network. Such failures might be expressed as the probability that a certain functional of a stochastic process exceeds a high threshold. In this talk I will present the framework of regular variation for stochastic processes and show how it can be applied to derive asymptotic approximations of failure probabilties. We are specifically interested in explaining what the typcial extreme sample paths of heavytailed stochastic processes looks like. The idea is to describe the extremal behavior of the process in terms of a limiting measure. Then a mapping theorem can be applied to obtain the asymptotic decay of functionals of the sample paths of the process. We will cover heavytailed Lévy processes, filtered Lévy processes, stochastic integrals, as well as large deviations for multivariate random walks with regularly varying steps. Towards the end I will also discuss an application in insurance.
[Refreshments will be served at 3:45 p.m. in Room 008 of the Statistics Building]
