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

By Jean Bertoin, Marc Yor. 2001

95 citations recorded by the source corpus.

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 $\xi$.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/90356b62b782f4fecb7dc1aa1e3eb61155705052

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In Mathematical Physics

Citations in the corpus

On the Lamperti stable processes (2008, background): “- There is another example in [4] which is related to the factorization”
Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes (2010, background): “ conditioned to stay positive (see the original paper [23] by Lamperti for the deﬁnition of this transformation and its various properties). Spectrally one-sided Lamperti-stable processes appeared in [8] and [27], and two-sided processes were studied in [11, 12, 14] and [22]. Lamperti-stable process”
Asymptotic laws for compositions derived from transformed subordinators (2004, background): “is the widely studied exponential functional, see [4, 5, 7, 28, 29].”
Lamperti-type laws (2007, result): “2007/02/20 file: ArLamperti.tex date: August 31, 2009 [see also Pitman (77)]for applications to Bayesian statistics, where in particular Pα,θ is referred to as a Pitman-Yor process.”
Regenerative composition structures (2003, background): “Extensive discussion of the exponential functional ∫∞ 0 exp(−St) dt is found in [6, 21].”