Importance sampling techniques for the multidimensional ruin problem for general Markov additive sequences of random vectors

Abstract

Let {(Xn, Sn): n = 0,1,...} be a Markov additive process, where {Xn} is a Markov chain on a general state space and Sn is an additive component on ℝd. We consider P{Sn ∈ A/ε, some n} as ε → 0, where A ⊂ ℝd is open and the mean drift of {Sn} is away from A. Our main objective is to study the simulation of P{Sn ∈ A/ε, some n) using the Monte Carlo technique of importance sampling. If the set A is convex, then we establish (i) the precise dependence (as ε → 0) of the estimator variance on the choice of the simulation distribution and (ii) the existence of a unique simulation distribution which is efficient and optimal in the asymptotic sense of D. Siegmund [Ann. Statist. 4 (1976) 673-684]. We then extend our techniques to the case where A is not convex. Our results lead to positive conclusions which complement the multidimensional counterexamples of P. Glasserman and Y. Wang [Ann. Appl. Probab. 7 (1997) 731-746].