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On subordinators, self-similar Markov processes and some factorizations of the exponential variable. (English) Zbl 1024.60030
From the authors’ abstract: Let \(\xi\) be a subordinator with Laplace exponent \(\Phi\), \[ I=\int^\infty_0 \exp(-\xi_s)ds \] the so-called exponential functional, and \(X\) (respectively, \(\widehat X)\) the self-similar Markov process obtained from \(\xi\) (respectively, from \(\widehat\xi= -\xi)\) by Lamperti’s transformation. The authors establish the existence of a unique probability measure \(\rho\) on \(]0,\infty[\) with \(k\)th moment given for every \(k\in \mathbb{N}\) by the product \(\Phi(1) \dots \Phi(k)\), and which bears some remarkable connections with the preceding variables. In particular, they show that if \(R\) is an independent random variable with law \(\rho\), then \(IR\) is a standard exponential variable, that the function \(t\to\mathbb{E} (1/X_t)\) coincides with the Laplace transform of \(\rho\), and that \(\rho\) is the 1-invariant distribution of the sub-Markovian process \(X\). A number of known factorizations of an exponential variable are shown to be of the preceding form \(IR\) for various subordinators \(\xi\).

MSC:
60J25 Continuous-time Markov processes on general state spaces
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