Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### On Dufresne's perpetuity, translated and reflected

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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.