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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

- M. BERTOIN
**M. YOR**

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

- 60J30 Processes with independent increments

**Résumé:** Let $\xi$ be a subordinator with Laplace
exponent $\Phi$, $I=\int_{0}^{\infty}\exp(-\xi_s)ds$ the so-called
exponential functional, and $X$ (respectively, $\hat X$) the
self-similar Markov
process obtained from $\xi$ (respectively, from $\hat{\xi}=-\xi$) by
Lamperti's
transformation. We establish the existence of a unique probability
measure
$\rho$ on $]0,\infty[$ with $k$-th moment given for
every $k\in\N$ by the product $\Phi(1)\cdots\Phi(k)$, and
which bears some remarkable
connections with the preceding variables. In particular we show that if
$R$ is an independent
random variable with law
$\rho$ then $IR$ is a standard exponential variable, that the function
$t\to\E(1/X_t)$ coincides with the Laplace transform of
$\rho$, and that $\rho$ is the $1$-invariant distribution of the
sub-markovian
process $\hat X$.

**Mots Clés:** *Self-similar Markov process ; subordinator ; exponential functional*

**Date:** 2001-06-11

**Prépublication numéro:** *PMA-666*