## Electronic Journal of Probability

### Extended factorizations of exponential functionals of Lévy processes

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 38, 22 pp.

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

https://projecteuclid.org/euclid.ejp/1465062360

Digital Object Identifier
doi:10.1214/EJP.v17-2057

Mathematical Reviews number (MathSciNet)
MR2928721

Zentralblatt MATH identifier
1253.60063

Rights

#### Citation

Patie, Pierre; Savov, Mladen. Extended factorizations of exponential functionals of Lévy processes. Electron. J. Probab. 17 (2012), paper no. 38, 22 pp. doi:10.1214/EJP.v17-2057. https://projecteuclid.org/euclid.ejp/1465062360

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