Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

On Dufresne's perpetuity, translated and reflected


Code(s) de Classification MSC:

Résumé: Let $B^{(\mu)}$ denote a Brownian motion with drift $\mu.$ In this paper we study two perpetual integral functionals of $B^{(\mu)}.$ The first one, introduced and investigated by Dufresne in \cite{dufresne90}, is $$ \int_0^\infty \exp(2\,B^{(\mu)}_s)\,ds,\quad \mu<0. $$ %and was introduced by Dufresne. It is known that this functional is identical in law with the first hitting time of 0 for a Bessel process with index $\mu.$ In particular, we analyze the following reflected (or one-sided) variants of Dufresne's functional $$ \int_0^{\infty} \exp(2\,B^{(\mu)}_s)\, {\bf 1}_{\{B^{(\mu)}_s> 0\}}\, ds, $$ and $$ %\quad{\rm and}\quad \int_0^{\infty} \exp(2\,B^{(\mu)}_s)\, {\bf 1}_{\{B^{(\mu)}_s< 0\}}\, ds. $$ These functionals can also be connected to hitting times. Our second functional, which we call Du\-fresne's translated functional, is $$ D_\nu:=\int_0^{\infty} (c+\exp(B^{(\nu)}_s))^{-2}\, ds, $$ where $c$ and $\nu$ are positive. This functional has all its moments finite, in contrast to Dufresne's functional which has only some finite moments. We compute explicitly the Laplace transform of $D_\nu$ in the case $\nu=1/2$ (other parameter values do not seem to allow explicit solutions) and connect this variable, as well as its reflected variants, to hitting times.

Mots Clés: Geometric Brownian motion ; Brownian motion with drift ; Bessel process ; Lamperti's transformation ; local time ; hitting times ; occupation times

Date: 2003-11-04

Prépublication numéro: PMA-858