MaPhySto
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

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Funded by The Danish National Research Foundation

Seminar

Abstract
The interest in heavy tails has had a boom in the last decade in application areas such as insurance risk, finance and networks engineering. The emphasis is often on studying the influence of the tail on a small probability, say the probability of a large financial loss or of incorrect transmission of a package in a communications network. Simulation is often the only possibility, but how to perform it efficiently is far less understood than in the light tailed case where the typical approach is importance sampling using i.i.d. exponential change of measure. A few efficient algorithms have been developed in the (overly simple) setting of $P(S_n>u)$, with $S_n=Y_1+cdots+Y_n$ a random walk and $n$ deterministic or an independent r.v. One of these uses conditional Monte Carlo and order statistics, others i.i.d. importance sampling. In this talk, I will show first how some of the importance sampling algorithms may be understood in terms of a minimum entropy (maximum likelihood) argument, and next how the identity $P(S_n>u)=$ $nP(S_n>u,M_n=Y_n)$, where $M_n=max(Y_1,ldots,Y_n)$, may be exploited to develop new and extremely efficient algorithms. Much of the intuition behind the whole talk is based upon $S_n$ and $M_n$ being close in the heavy-tailed case.

Contact person:Søren Asmussen.