Importance sampling for stochastic recurrence equations with heavy tailed increments

Importance sampling in the setting of heavy tailed random variables has generally focused on models with additive noise terms. In this work we extend this concept by considering importance sampling for the estimation of rare events in Markov chains of the form X _n+1 = A _n+1X _n+B _n+1, X ₀ = 0, where the B _n's and A _n's are independent sequences of independent and identically distributed (i.i.d.) random variables and the B _n's are regularly varying and the A _n's are suitably light tailed relative to B _n. We focus on efficient estimation of the rare event probability P(X _n > b) as b↗∞. In particular we present a strongly efficient importance sampling algorithm for estimating these probabilities, and present a numerical example showcasing the strong efficiency.