## Bernoulli

• Bernoulli
• Volume 19, Number 5A (2013), 1938-1964.

### On the density of exponential functionals of Lévy processes

#### Abstract

In this paper, we study the existence of the density associated with the exponential functional of the Lévy process $\xi$,

$I_{\mathbf{e} _{q}}:=\int_{0}^{\mathbf{e} _{q}}\mathrm{e}^{\xi_{s}}\,\mathrm{d}s,$

where $\mathbf{e} _{q}$ is an independent exponential r.v. with parameter $q\geq0$. In the case where $\xi$ is the negative of a subordinator, we prove that the density of $I_{\mathbf{e}_{q}}$, here denoted by $k$, satisfies an integral equation that generalizes that reported by Carmona et al. [7]. Finally, when $q=0$, we describe explicitly the asymptotic behavior at $0$ of the density $k$ when $\xi$ is the negative of a subordinator and at $\infty$ when $\xi$ is a spectrally positive Lévy process that drifts to $+\infty$.

#### Article information

Source
Bernoulli, Volume 19, Number 5A (2013), 1938-1964.

Dates
First available in Project Euclid: 5 November 2013

https://projecteuclid.org/euclid.bj/1383661209

Digital Object Identifier
doi:10.3150/12-BEJ436

Mathematical Reviews number (MathSciNet)
MR3129040

Zentralblatt MATH identifier
1305.60035

#### Citation

Pardo, J.C.; Rivero, V.; van Schaik, K. On the density of exponential functionals of Lévy processes. Bernoulli 19 (2013), no. 5A, 1938--1964. doi:10.3150/12-BEJ436. https://projecteuclid.org/euclid.bj/1383661209

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