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

### Perpetual integral functionals as hitting and occupation times

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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.