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2012 Extended factorizations of exponential functionals of Lévy processes
Pierre Patie, Mladen Savov
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Electron. J. Probab. 17: 1-22 (2012). DOI: 10.1214/EJP.v17-2057


In [18], under mild conditions, a Wiener-Hopf type factorization is derived for the exponential functional of proper Lévy processes. In this paper, we extend this factorization by relaxing a finite moment assumption as well as by considering the exponential functional for killed Lévy processes. As a by-product, we derive some interesting fine distributional properties enjoyed by a large class of this random variable, such as the absolute continuity of its distribution and the smoothness, boundedness or complete monotonicity of its density. This type of results is then used to derive similar properties for the law of maxima and first passage time of some stable Lévy processes. Thus, for example, we show that for any stable process with $\rho \in (0,\frac{1}{\alpha} - 1]$ where $\rho \in [0,1]$ is the positivity parameter and $\alpha$ is the stable index, then the first passage time has a bounded and non-increasing density on $\mathbb{R}_+$. We also generate many instances of integral or power series representations for the law of the exponential functional of Lévy processes with one or two-sided jumps. The proof of our main results requires different devices from the one developed in [18]. It relies in particular on a generalization of a transform recently introduced in [9] together with some extensions to killed Lévy process of Wiener-Hopf techniques. The factorizations developed here also allow for further applications which we only indicate here.


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Pierre Patie. Mladen Savov. "Extended factorizations of exponential functionals of Lévy processes." Electron. J. Probab. 17 1 - 22, 2012.


Accepted: 30 May 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1253.60063
MathSciNet: MR2928721
Digital Object Identifier: 10.1214/EJP.v17-2057

Primary: 60G51
Secondary: 44A60 , 60E07 , 60J55

Keywords: complete monotonicity , exponential functional , Infinite divisibility , Lévy processes , Special functions , Stable Lévy processes , Wiener-Hopf factorizations

Vol.17 • 2012
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