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

   
@article{ECP1039,
	author = {Jean Bertoin and Marc Yor},
	title = {On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {6},
	year = {2001},
	keywords = {Self-similar Markov process, subordinator, exponential functional},
	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 $\xi$.},
	pages = {no. 10, 95-106},
	issn = {1083-589X},
	doi = {10.1214/ECP.v6-1039},    
        url = {http://ecp.ejpecp.org/article/view/1039}}