## Journal of Applied Probability

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

### On exact sampling of stochastic perpetuities

Jose H. Blanchet and Karl Sigman

#### Abstract

A stochastic perpetuity takes the form *D*∞=∑_{n=0}^{∞}
exp(*Y*_{1}+⋯+*Y*_{n})*B*
_{n}, 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}=e^{Yn}; *D*_{∞} then satisfies
the stochastic fixed-point equation *D*_{∞}D̳*AD*_{∞}+*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 E*B*_{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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 18 October 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1318940463

**Digital Object Identifier**

doi:10.1239/jap/1318940463

**Mathematical Reviews number (MathSciNet)**

MR2865624

**Zentralblatt MATH identifier**

1230.65012

**Subjects**

Primary: 65C05: Monte Carlo methods

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

**Keywords**

Perfect sampling coupling from the past Markov chain stochastic perpetuity

#### Citation

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