On the Range of Exponential Functionals of Lévy Processes

Abstract

We characterize the support of the law of the exponential functional \(\int _{0}^{\infty }e^{-\xi _{s-}}\,d\eta _{s}\) of two one-dimensional independent Levy processes ξ and η. Further, we study the range of the mapping Φ ξ for a fixed Levy process ξ, which maps the law of η1 to the law of the corresponding exponential functional \(\int _{0}^{\infty }e^{-\xi _{s-}}\,d\eta _{s}\). It is shown that the range of this mapping is closed under weak convergence and in the special case of positive distributions several characterizations of laws in the range are given.