## Electronic Journal of Probability

### Distributional properties of exponential functionals of Lévy processes

#### Abstract

We study the distribution of the exponential functional $I(\xi,\eta)=\int_0^{\infty} \exp(\xi_{t-}) d \eta_t$, where $\xi$ and $\eta$ are independent Lévy processes. In the general setting, using the theory of Markov processes and Schwartz distributions, we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in \cite{CPY}. In the special case when $\eta$ is a Brownian motion  with drift, we show that this integral equation leads to an important functional equation for the Mellin transform of $I(\xi,\eta)$, which proves to be a very useful tool for studying the distributional properties of this random variable. For general Lévy process $\xi$ ($\eta$ being Brownian motion with drift) we prove that the exponential functional has a smooth density on $\mathbb{R} \setminus \{0\}$, but surprisingly the second derivative at zero may fail to exist. Under the additional assumption that $\xi$ has some positive exponential moments we establish an asymptotic behaviour of $\mathbb{P}(I(\xi,\eta)>x)$ as $x\to +\infty$,  and under similar assumptions on the negative exponential moments of $\xi$ we obtain a precise asymptotic expansion of the density of $I(\xi,\eta)$ as $x\to 0$. Under further assumptions on the Lévy process $\xi$ one is able to prove much stronger results about the  density of the exponential functional and we illustrate some of the ideas and techniques for the case when $\xi$ has hyper-exponential jumps.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 8, 35 pp.

Dates
Accepted: 25 January 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062330

Digital Object Identifier
doi:10.1214/EJP.v17-1755

Mathematical Reviews number (MathSciNet)
MR2878787

Zentralblatt MATH identifier
1246.60073

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes

Rights

#### Citation

Kuznetsov, Alexey; Pardo, Juan Carlos; Savov, Mladen. Distributional properties of exponential functionals of Lévy processes. Electron. J. Probab. 17 (2012), paper no. 8, 35 pp. doi:10.1214/EJP.v17-1755. https://projecteuclid.org/euclid.ejp/1465062330

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