On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable

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 $k\in N$ 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 $\hat X$. A number of known factorizations of an exponential variable are shown to be of the preceding form $IR$ for various subordinators $\xi$.