The Annals of Applied Probability

Rare event simulation for processes generated via stochastic fixed point equations

Jeffrey F. Collamore, Guoqing Diao, and Anand N. Vidyashankar

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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.

Article information

Ann. Appl. Probab., Volume 24, Number 5 (2014), 2143-2175.

First available in Project Euclid: 26 June 2014

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Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods 91G60: Numerical methods (including Monte Carlo methods) 68W40: Analysis of algorithms [See also 68Q25] 60H25: Random operators and equations [See also 47B80]
Secondary: 60F10: Large deviations 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J05: Discrete-time Markov processes on general state spaces 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J22: Computational methods in Markov chains [See also 65C40] 60K15: Markov renewal processes, semi-Markov processes 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx] 60G70: Extreme value theory; extremal processes 68U20: Simulation [See also 65Cxx] 91B30: Risk theory, insurance 91B70: Stochastic models 91G70: Statistical methods, econometrics

Monte Carlo methods importance sampling perpetuities large deviations nonlinear renewal theory Harris recurrent Markov chains first entrance times last exit times regeneration times financial time series GARCH processes ARCH processes risk theory ruin theory with stochastic investments


Collamore, Jeffrey F.; Diao, Guoqing; Vidyashankar, Anand N. Rare event simulation for processes generated via stochastic fixed point equations. Ann. Appl. Probab. 24 (2014), no. 5, 2143--2175. doi:10.1214/13-AAP974.

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