Exponential Functionals of Lévy Processes with Jumps

Alea, Lat
2015 Am. J. Probab. Math. Stat   unpublished
We study the exponential functional ∫ ∞ 0 e −ξs− dη s of two one-dimensional independent Lévy processes ξ and η, where η is a subordinator. In particular, we derive an integro-differential equation for the density of the exponential functional whenever it exists. Further, we consider the mapping Φ ξ for a fixed Lévy process ξ, which maps the law of η 1 to the law of the corresponding exponential functional ∫ ∞ 0 e −ξs− dη s , and study the behaviour of the range of Φ ξ for varying
more » ... s of ξ. Moreover, we derive conditions for selfdecomposable distributions and generalized Gamma convolutions to be in the range. On the way we also obtain new characterizations of these classes of distributions.