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October 2014 Rare event simulation for processes generated via stochastic fixed point equations
Jeffrey F. Collamore, Guoqing Diao, Anand N. Vidyashankar
Ann. Appl. Probab. 24(5): 2143-2175 (October 2014). DOI: 10.1214/13-AAP974

Abstract

In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable $V$ satisfying the distributional equation $V\mathop{=}^\mathcal{D}f(V)$, where $f(v)=A\max\{v,D\}+B$ for $(A,B,D)\in(0,\infty)\times\mathbb{R}^{2}$. This paper is concerned with computational methods for evaluating these tail probabilities. We introduce a novel importance sampling algorithm, involving an exponential shift over a random time interval, for estimating these rare event probabilities. We prove that the proposed estimator is: (i) consistent, (ii) strongly efficient and (iii) optimal within a wide class of dynamic importance sampling estimators. Moreover, using extensions of ideas from nonlinear renewal theory, we provide a precise description of the running time of the algorithm. To establish these results, we develop new techniques concerning the convergence of moments of stopped perpetuity sequences, and the first entrance and last exit times of associated Markov chains on $\mathbb{R}$. We illustrate our methods with a variety of numerical examples which demonstrate the ease and scope of the implementation.

Citation

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Jeffrey F. Collamore. Guoqing Diao. Anand N. Vidyashankar. "Rare event simulation for processes generated via stochastic fixed point equations." Ann. Appl. Probab. 24 (5) 2143 - 2175, October 2014. https://doi.org/10.1214/13-AAP974

Information

Published: October 2014
First available in Project Euclid: 26 June 2014

zbMATH: 1316.65015
MathSciNet: MR3226174
Digital Object Identifier: 10.1214/13-AAP974

Subjects:
Primary: 60H25, 65C05, 68W40, 91G60

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 5 • October 2014
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