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]
- 60J70 Applications of diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

**Résumé:** Let $X$ be a linear diffusion and $f$ a non-negative, Borel
measurable function. We are interested in finding conditions
on $X$ and $f$ which imply that %and explicit examples
the perpetual integral functional
$$
I^X_\infty(f):=\int_0^\infty f(X_t)\, dt
$$
is identical in law with the first hitting time of a point for some other
diffusion. This phenomenon may often be explained using random time
change. Because of some potential
applications in mathematical finance, we are considering mainly the case
when $X$ is a Brownian motion with drift $\mu>0,$ denoted
$\{B^{(\mu)}_t:\ t\geq 0\},$ but it is obvious
that the method presented is more general. We also review the
known examples and give new ones. In particular, results concerning one-sided
functionals
$$
\int_0^\infty f(B^{(\mu)}_t)\,{\bf 1}_{\{B^{(\mu)}_t<0\}} dt\quad
{\rm and}\quad
\int_0^\infty f(B^{(\mu)}_t)\,{\bf 1}_{\{B^{(\mu)}_t>0\}} dt
$$
are presented.
This approach
generalizes the proof, based on the random time change techniques,
of the fact that the Dufresne functional (this
corresponds to $f(x)=\exp(-2x)),$ playing quite an important
r\^ole in the study of geometric Brownian motion, is identical in law with the
first hitting time for a Bessel process. Another functional
arising naturally in this context is %associated to the function
%$f(x)=(a+\exp(x))^{-2},\ a>0.$
$$
\int_0^\infty \big(a+\exp(B^{(\mu)}_t)\big)^{-2}\, dt,
$$
which is seen, in the case $\mu=1/2,$ to be identical in
law with the first hitting time for a Brownian motion with drift
$\mu=a/2.$
The paper is concluded by discussing how the
Feynman-Kac formula can be used to find the distribution of a
perpetual integral functional.

**Mots Clés:** *Time change ; Lamperti transformation ; Bessel processes ;
Ray-Knight theorems ; feynman-Kac formula*

**Date:** 2004-03-02

**Prépublication numéro:** *PMA-887*