On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable
Jean Bertoin, Universite Pierre et Marie Curie
Marc Yor, Universite Pierre et Marie Curie
Abstract
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 $kinN$ by the product $Phi(1)cdotsPhi(k)$, 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
$ttoE(1/X_t)$ coincides with the Laplace transform of
$rho$, and that $rho$ is the $1$-invariant distribution of the sub-markovian
process $hat X$.
A number of known factorizations of an exponential variable are shown to be of the preceding
form $IR$ for various subordinators $xi$.
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