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

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

- P. SALMINEN
**M. YOR**

**Code(s) de Classification MSC:**

- 60J65 Brownian motion [See also 58J65]
- 60J60 Diffusion processes [See also 58J65]
- 60J55 Local time and additive functionals
- 91B30 Risk theory, insurance

**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*