Hostname: page-component-7f64f4797f-fz5kh Total loading time: 0 Render date: 2025-11-08T18:20:38.565Z Has data issue: false hasContentIssue false
  • English
  • Français

On exact sampling of stochastic perpetuities

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Jose H. Blanchet  and
Karl Sigman
Jose H. Blanchet
Affiliation:
Columbia University, Department of Industrial Engineering and Operations Research, Columbia University, S.W. Mudd Building, 500 West 120th Street, New York, NY 10025, USA
Karl Sigman
Affiliation:
Columbia University, Department of Industrial Engineering and Operations Research, Columbia University, S.W. Mudd Building, 500 West 120th Street, New York, NY 10025, USA. Email address: karl.sigman@columbia.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A stochastic perpetuity takes the form D∞=∑n=0 ∞ exp(Y 1+⋯+Y n )Bn , where Y n :n≥0) and (B n :n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively by D n+1=A n D n +B n , n≥0, where A n =eY n ; D ∞ then satisfies the stochastic fixed-point equation D ∞D̳AD ∞+B, where A and B are independent copies of the A n and B n (and independent of D ∞ on the right-hand side). In our framework, the quantity B n , which represents a random reward at time n, is assumed to be positive, unbounded with EB n p <∞ for some p>0, and have a suitably regular continuous positive density. The quantity Y n is assumed to be light tailed and represents a discount rate from time n to n-1. The RV D ∞ then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples of D ∞. Our method is a variation of dominated coupling from the past and it involves constructing a sequence of dominating processes.

Keywords

Perfect samplingcoupling from the pastMarkov chainstochastic perpetuity

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60J05: Discrete-time Markov processes on general state spaces 68U20: Simulation

Information

Type
Part 4. Simulation
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 165 - 182
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Aldous, D. J. and Bandyopadhyay, A., (2005). A survey of max-type recursive distributional equations. Ann. Appl. Prob. 15, 1047–1110.Google Scholar
[2] Asmussen, S. and Glynn, P. W., (2008). Stochastic Simulation: Algorithms and Analysis. Springer, New York.Google Scholar
[3] Campbell, J. Y., Lo, A. W. and MacKinlay, C., (1999). The Econometrics of Financial Markets. Princeton University Press, Princeton, NJ.Google Scholar
[4] Carmona, P., Petit, F. and Yor, M., (2001). Exponential functionals of L{é}vy processes. In L{é}vy Processes}, eds Barndorff-Nielsen, O. et al., Birkhäuser, Boston, MA, pp. 41–55.CrossRefGoogle Scholar
[5] Connor, S. B. and Kendall, W. S., (2007). Perfect simulation for a class of positive recurrent Markov chains. Ann. Appl. Prob. 3, 781–808.Google Scholar
[6] Devroye, L., (2001). Simulating perpetuities. Methodology Comput. Appl. Prob. 3, 97–115.Google Scholar
[7] Devroye, L. and Neininger, R., (2002). Density approximation and exact simulation of random variables that are the solutions of fixed-point equations. Adv. Appl. Prob. 34, 441–468.Google Scholar
[8] Dufresne, D., (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 1990, 39–79.Google Scholar
[9] Embrechts, P. and Goldie, C. M., (1994). Perpetuities and random equations. In Asymptotic Statistics (Proc. 5th Prague Symp.), Physica, Heidelberg, pp. 75–86.Google Scholar
[10] Ensor, K. B. and Glynn, P. W., (2000). Simulating the maximum of a random walk. J. Statist. Planning Infer. 85, 127–135.Google Scholar
[11] Fill, J. A. and Huber, M. L., (2010). Perfect simulation of Vervaat perpetuities. Electron. J. Prob. 15, 96–109.Google Scholar
[12] Gjessing, H. K. and Paulsen, J., (1997). Present value distributions with applications to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123–144.CrossRefGoogle Scholar
[13] Goldie, C. M. and Grübel, R., (1996). Perpetuities with thin tails. Adv. Appl. Prob. 28, 463–480.Google Scholar
[14] Harrison, J. M., (1977). Ruin problems with compounding assets. Stoch. Process. Appl. 5, 67–79.Google Scholar
[15] Jelenkovic, P. R. and Olvera-Cravioto, M., (2010). Implicit renewal theory and power tails on trees. Preprint. Available at http://arxiv.org/abs/1006.3295v3.Google Scholar
[16] Kella, O., (2009). On growth-collapse processes with stationary structure and their shot-noise counterparts. J. Appl. Prob. 46, 363–371.CrossRefGoogle Scholar
[17] Kendall, W. S., (2004). Geometric ergodicity and perfect simulation. Electron. Commun. Prob. 9, 140–151.Google Scholar
[18] Kesten, H., (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.Google Scholar
[19] Paulsen, J., (1998). Sharp conditions for certain ruin in a risk process with stochastic return on investments. Stoch. Process. Appl. 75, 135–148.CrossRefGoogle Scholar
[20] Pollack, M. and Siegmund, D., (1985). A diffusion and its applications to detecting a change in the drift of Brownian motion. Biometrika 72, 267–280.Google Scholar
[21] Propp, J. G. and Wilson, D. B., (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.3.0.CO;2-O>CrossRefGoogle Scholar
[22] Maulik, K. and Zwart, B., (2006). Tail asymptotics of exponential functionals of L{é}vy processes. Stoch. Process. Appl. 116, 156–177.Google Scholar
[23] Nyrhinen, H., (2001). Finite and infinite time ruin probabilities in a stochastic economic environment. Stoch. Process. Appl. 92, 265–285.Google Scholar
[24] Vervaat, W., (1977). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar