On the density of exponential functionals of Lévy processes

In this paper, we study the existence of the density associated with the exponential functional of the Lévy process ξ, (Equation presented) where eq is an independent exponential r.v. with parameter q ≥ 0. In the case where ξ is the negative of a subordinator, we prove that the density of Ieq, 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 ξ is the negative of a subordinator and at ∞ when ξ is a spectrally positive Lévy process that drifts to +∞. © 2013 ISI/BS.