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

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{'e}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.