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February 2002 Importance Sampling Techniques for the Multidimensional Ruin Problem for General Markov Additive Sequences of Random Vectors
J.F. Collamore
Ann. Appl. Probab. 12(1): 382-421 (February 2002). DOI: 10.1214/aoap/1015961169

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].

Citation

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J.F. Collamore. "Importance Sampling Techniques for the Multidimensional Ruin Problem for General Markov Additive Sequences of Random Vectors." Ann. Appl. Probab. 12 (1) 382 - 421, February 2002. https://doi.org/10.1214/aoap/1015961169

Information

Published: February 2002
First available in Project Euclid: 12 March 2002

zbMATH: 1021.65003
MathSciNet: MR1890070
Digital Object Identifier: 10.1214/aoap/1015961169

Subjects:
Primary: 65C05
Secondary: 60F10 , 60J15 , 60K10 , 65U05

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

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.12 • No. 1 • February 2002
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