Martingales, scale functions and stochastic life annuities: A note
Consider the process \(Y_t = \int_0^t \alpha(s) \exp(-\sigma W_s) ds\) where \(W\) is Brownian motion and \(\alpha\geq 0\) is a smooth function. This paper gives an expression for the distribution function of \(Y_\infty\) in terms of the solution \(F\) of a diffusion PDE involving \(\alpha\). For \(\alpha(s) = \exp(-\delta s)\), this yields a simple proof of a result due to \textit{D. Dufresne} [Scand. Actuarial J. 1990, No. 1/2, 39-79 (1990; Zbl 0743.62101)] on stochastic perpetuities. The idea behind the general result is to transform \((t,W_t, Y_t)\) into a Markov process \((U_t)\) and then find a function \(F\) such that \(F(t,U_t)\) is a local martingale.
scientific article; zbMATH DE number 1310307