Martingales, scale functions and stochastic life annuities: A note
type: Item
instance of: scholarly article
title: Martingales, scale functions and stochastic life annuities: A note
zbMATH Open document ID: 0930.60060
DOI: 10.1016/S0167-6687(98)00032-8
author: Moshe Arye Milevsky
published in: Insurance Mathematics \& Economics
publication date: 2000-01-31T00:00:00Z
review text: 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.
reviewed by: Martin Schweizer
Mathematics Subject Classification ID: 60J70
zbMATH DE Number: 1310307
zbMATH Keywords: stochastic perpetuities
zbMATH Keywords: martingales
zbMATH Keywords: scale function
MaRDI profile type: Publication
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full work available at URL: s0167-6687(98)00032-8
OpenAlex ID: W1984468925
description: scientific article; zbMATH DE number 1310307