The present value of a stochastic perpetuity and the gamma distribution. (English) Zbl 0903.60069

Summary: 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.


60J60 Diffusion processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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