Importance Sampling Techniques for the Multidimensional Ruin Problem for General Markov Additive Sequences of Random Vectors

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/.,some n} as ..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/.,some n} using the Monte Carlo technique of importance sampling. If the set A is convex, then we establish (i) the precise dependence (as ..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].