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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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