Exponential functionals of spectrally one-sided l{é}vy processes conditioned to stay positive

Abstract

We study the properties of the exponential functional $\int \_0^{+\infty }{e}^{-X^{\uparrow }(t)}dt$ where $X^{\uparrow }$ is a spectrally one-sided L{é}vy process conditioned to stay positive. In particular, we study finiteness, self-decomposability, existence of finite exponential moments, asymptotic tail at $0$ and smoothness of the density.