On exact sampling of stochastic perpetuities

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 Y n ; D ∞ then satisfies the stochastic fixed-point equation D ∞D̳AD ∞+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 EB 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.