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Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration. (English) Zbl 1171.60009
Let the exponential functional associated to a Levy process \(\xi \) with a negative first moment be defined as \(\int_0^{\infty} \exp\{\xi(s)\}ds\). Let \( (\lambda)_{\alpha} := \frac{\Gamma(\lambda + \alpha)}{\Gamma(\lambda)}\).
In this paper the author study a two \((\alpha , \gamma)\)-parameters family of spectrally negative Levy processes which Laplace exponents have the form \[ \psi^{(\gamma)}(\lambda) = c((\lambda + \gamma)_{\alpha} - (\gamma)_{\alpha}) , \quad \lambda \geq 0 \] with a positive constant \(c\). Furthermore, the author compute the density of the law of the exponential functional associated to Levy processes which admit the following Laplace exponent: for any \(\delta > \frac{\alpha -1}{\alpha}\) \[ \psi^{(0,\delta)}(\lambda) = \psi^{(0)}(\lambda) - \frac{\alpha\delta}{\lambda + \alpha -1}\psi^{(0)}(\lambda) , \quad \lambda \geq 0 . \] The path, he follows, goes through characterizing the family of Levy processes associated, via the Lamperti mapping, to the family of self-similar continuous state branching processes with immigration (briefly cbip). Using well-known results on cbip, the author derives the spatial Laplace transforms of the semi-group of this family and computes the corresponding densities.

MSC:
60G51 Processes with independent increments; Lévy processes
60E07 Infinitely divisible distributions; stable distributions
60G18 Self-similar stochastic processes
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
33E12 Mittag-Leffler functions and generalizations
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