Topics in rare events simulation

by Guiliana Torrisi

lectures in ENS, April, 2005

1     Introduction
       1.1   The problem of rare events simulation
     1.2   Asymptotically optimal estimators

2     Level crossing probabilities under light tail conditions
      2.1   Admissible and asymptotically optimal laws

       2.2   Random walks with negative drift

         2.2.1      Lundberg upper bound
         2.2.2      Exact asymptotics
         2.2.3      Cramer-Lundberg model
        2.2.4      Admissible and asymptotically optimal law
        2.2.5      Numerical illustrations
        2.2.6      Conditioned limit interpretation

      2.3   Poisson shot noise models

        2.3.1      Lundberg upper bound and asymptotics
        2.3.2      Admissible and asymptotically optimal law
        2.3.3      Numerical illustrations

3     Level crossing probabilities under heavy tail conditions

      3.1   Sums of random variables

         3.1.1      An algorithm involving order statistics
         3.1.2      Numerical illustrations
        3.1.3      Extension to random sums