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 Lévy processes $\xi$ and $\eta$. Further, we study the range of the mapping ${\Phi }_{\xi }$ for a fixed Lévy process $\xi$, which maps the law of ${\eta }_1$ 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.