Journal of Applied Probability

On exact sampling of stochastic perpetuities

Jose H. Blanchet and Karl Sigman


A stochastic perpetuity takes the form D∞=∑n=0 exp(Y1+⋯+Yn)B n, 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=An Dn+Bn, n≥0, where An=eYn; D then satisfies the stochastic fixed-point equation DAD+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.

Article information

J. Appl. Probab., Volume 48A (2011), 165-182.

First available in Project Euclid: 18 October 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Perfect sampling coupling from the past Markov chain stochastic perpetuity


Blanchet, Jose H.; Sigman, Karl. On exact sampling of stochastic perpetuities. J. Appl. Probab. 48A (2011), 165--182. doi:10.1239/jap/1318940463.

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