The present value of a stochastic perpetuity and the Gamma distribution

We derive the probability density function of the present value of a perpetuity subjected to a stochastic Wiener rate of interest and prove that its inverse is Gamma distributed. This result is useful for computing the initial endowment required to fund a perpetuity, in a real world stochastic environment, under a fixed probabilistic confidence level. The proof relies on well-known martingale results from the theory of stochastic calculus. A numerical example is provided with tables.