The Annals of Applied Probability

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

J.F. Collamore

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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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1015961169

Digital Object Identifier
doi:10.1214/aoap/1015961169

Mathematical Reviews number (MathSciNet)
MR1890070

Zentralblatt MATH identifier
1021.65003

Subjects
Primary: 65C05: Monte Carlo methods
Secondary: 65U05 60F10: Large deviations 60J15 60K10: Applications (reliability, demand theory, etc.)

Keywords
Monte Carlo methods rare event simulation hitting probabilities large deviations Harris recurrent Markov chains convex analysis

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