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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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.

Article information

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

Dates
First available in Project Euclid: 26 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1403812372

Digital Object Identifier
doi:10.1214/13-AAP974

Mathematical Reviews number (MathSciNet)
MR3226174

Zentralblatt MATH identifier
1316.65015

Subjects
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

Keywords
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

Citation

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. https://projecteuclid.org/euclid.aoap/1403812372


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References

  • Alsmeyer, G., Iksanov, A. and Rösler, U. (2009). On distributional properties of perpetuities. J. Theoret. Probab. 22 666–682.
  • Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.
  • Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.
  • Blanchet, J., Lam, H. and Zwart, B. (2012). Efficient rare-event simulation for perpetuities. Stochastic Process. Appl. 122 3361–3392.
  • Collamore, J. F. (2002). Importance sampling techniques for the multidimensional ruin problem for general Markov additive sequences of random vectors. Ann. Appl. Probab. 12 382–421.
  • Collamore, J. F. (2009). Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. Ann. Appl. Probab. 19 1404–1458.
  • Collamore, J. F. and Vidyashankar, A. N. (2013a). Large deviation tail estimates and related limit laws for stochastic fixed point equations. In Random Matrices and Iterated Random Functions (G. Alsmeyer and M. Löwe, eds.) 91–117. Springer, Heidelberg.
  • Collamore, J. F. and Vidyashankar, A. N. (2013b). Tail estimates for stochastic fixed point equations via nonlinear renewal theory. Stochastic Process. Appl. 123 3378–3429.
  • Collamore, J. F., Vidyashankar, A. N. and Xu, J. (2013). Rare event simulation for stochastic fixed point equations related to the smoothing transform. In Proceedings of the Winter Simulation Conference 555–563.
  • Dupuis, P. and Wang, H. (2005). Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Probab. 15 1–38.
  • Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126–166.
  • Heyde, C. C. (1966). Some renewal theorems with application to a first passage problem. Ann. Math. Statist. 37 699–710.
  • Iscoe, I., Ney, P. and Nummelin, E. (1985). Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math. 6 373–412.
  • Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.
  • Lai, T. L. and Siegmund, D. (1979). A nonlinear renewal theory with applications to sequential analysis. II. Ann. Statist. 7 60–76.
  • Letac, G. (1986). A contraction principle for certain Markov chains and its applications. In Random Matrices and Their Applications (Brunswick, Maine, 1984). Contemp. Math. 50 263–273. Amer. Math. Soc., Providence, RI.
  • Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Univ. Press, Cambridge.
  • Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673–684.
  • Siegmund, D. (1985). Sequential Analysis: Tests and Confidence Intervals. Springer, New York.
  • Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750–783.
  • Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia, PA.