Rare-event simulation for stochastic recurrence equations with heavy-tailed innovations

Abstract

In this article, rare-event simulation for stochastic recurrence equations of the form X n +1 = A n +1 X n + B n +1 , X 0 =0 is studied, where { A n ; n ≥ 1} and { B n ; n ≥ 1} are independent sequences consisting of independent and identically distributed real-valued random variables. It is assumed that the tail of the distribution of B 1 is regularly varying, whereas the distribution of A 1 has a suitably light tail. The problem of efficient estimation, via simulation, of quantities such as P { X n >b} and P {sup k ≤ n X k > b} for large b and n is studied. Importance sampling strategies are investigated that provide unbiased estimators with bounded relative error as b and n tend to infinity.