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On exact sampling of stochastic perpetuities

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

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
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Abstract

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A stochastic perpetuity takes the form D∞=∑n=0∞ exp(Y1+⋯+Yn)Bn, where Yn:n≥0) and (Bn: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 Dn+1=AnDn+Bn, n≥0, where An=eYn; D∞ then satisfies the stochastic fixed-point equation D∞D̳AD∞+B, where A and B are independent copies of the An and Bn (and independent of D∞ on the right-hand side). In our framework, the quantity Bn, which represents a random reward at time n, is assumed to be positive, unbounded with EBnp <∞ for some p>0, and have a suitably regular continuous positive density. The quantity Yn 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
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 

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