## The Annals of Applied Probability

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

J.F. Collamore

#### 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].

#### Article information

Source
Ann. Appl. Probab., Volume 12, Number 1 (2002), 382-421.

Dates
First available in Project Euclid: 12 March 2002

https://projecteuclid.org/euclid.aoap/1015961169

Digital Object Identifier
doi:10.1214/aoap/1015961169

Mathematical Reviews number (MathSciNet)
MR1890070

Zentralblatt MATH identifier
1021.65003

#### Citation

Collamore, J.F. Importance Sampling Techniques for the Multidimensional Ruin Problem for General Markov Additive Sequences of Random Vectors. Ann. Appl. Probab. 12 (2002), no. 1, 382--421. doi:10.1214/aoap/1015961169. https://projecteuclid.org/euclid.aoap/1015961169

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• ETH ZENTRUM, HG F 42.2 CH-8092 ZÜRICH SWITZERLAND E-MAIL: collamore@math.ethz.ch