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

Abstract

Let $\{(X_n, S_n): n = 0, 1, \dots\}$ be a Markov additive process, where $\{X_n\}$ is a Markov chain on a general state space and $S_n$ is an additive component on $\mathbb{R}^d$. We consider $\mathbf{P}\{S_n \in A/\varepsilon, \text{some $n$}\}$ as $\varepsilon \to 0$, where $A \subset \mathbb{R}^d$ is open and the mean drift of $\{S_n\}$ is away from $A$. Our main objective is to study the simulation of $\mathbf{P}\{S_n \in A/\varepsilon, \text{some $n$}\}$ using the Monte Carlo technique of importance sampling. If the set $A$ is convex, then we establish (i) the precise dependence (as $\varepsilon \to 0$) 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].