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 R . We consider P {Sn ∈ A/ , some n} as → 0, where A ⊂ R 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; (ii) the existence of a unique simulation distribution which is efficient and optimal in the asymptotic sense of Siegmund (1976). 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 Glasserman and Wang (1997).