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On exact sampling of stochastic perpetuities. (English) Zbl 1230.65012
A stochastic perpetuity takes the form $$D_\infty =\sum _{n=0}^{\infty } \exp(Y_{1}+ \cdots +Y_{n})B_{n}$$, where $$(Y_{n}:n\geq 0$$) and ($$B_{n}:n\geq 0$$) 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 $$D_{n+1}=A_{n} D_{n}+B_{n}$$, $$n\geq 0$$, where $$A_{n}=e^{Y_{n}}$$; $$D_\infty$$ then satisfies the stochastic fixed-point equation $$D_{\infty } \overset {D} = AD_{\infty }+B$$, where $$A$$ and $$B$$ are independent copies of the $$A_{n}$$ and $$B_{n}$$ (and independent of $$D_\infty$$ on the right-hand side).
The quantity $$B_{n}$$, which represents a random reward at time $$n$$, is assumed to be positive, unbounded with E$$B_{n}^{p} <\infty$$ 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 $$D_\infty$$ then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. The authors provide an exact simulation algorithm for generating samples of $$D_\infty$$. Their method is a variation of dominated coupling from the past and it involves constructing a sequence of dominating processes.

##### MSC:
 65C40 Numerical analysis or methods applied to Markov chains 60J05 Discrete-time Markov processes on general state spaces 60J22 Computational methods in Markov chains 62D05 Sampling theory, sample surveys
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