Martingales, scale functions and stochastic life annuities: a note

In this note we derive the most general conditions under which the probability distribution of a generalized stochastic life annuity can be obtained by using the scale function methodology. Our main result is that the cumulative distribution function (CDF) of the generalized stochastic life annuity will obey the partial differential equation (PDE) satisfied by the scale function whenever the underlying process can be “Markovianized”. The scale function is the mapping which converts a Markov diffusion process into a martingale. In many cases, the resulting PDE can be easily solved to yield a closed form expression for the CDF.