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2012 Distributional properties of exponential functionals of Lévy processes
Alexey Kuznetsov, Juan Carlos Pardo, Mladen Savov
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Electron. J. Probab. 17: 1-35 (2012). DOI: 10.1214/EJP.v17-1755

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.

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Alexey Kuznetsov. Juan Carlos Pardo. Mladen Savov. "Distributional properties of exponential functionals of Lévy processes." Electron. J. Probab. 17 1 - 35, 2012. https://doi.org/10.1214/EJP.v17-1755

Information

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

zbMATH: 1246.60073
MathSciNet: MR2878787
Digital Object Identifier: 10.1214/EJP.v17-1755

Subjects:
Primary: 60G51

Keywords: asymptotic expansions , exponential functional , integral equations , Lévy processes , Mellin transform

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