On exponential functionals of processes with independent increments. (arXiv:1610.08732v1 [math.PR])

In this paper we study the exponential functionals of the processes $X$ with independent increments , namely$$I\_t= \int \_0^t\exp(-X\_s)ds, \_,\,\, t\geq 0,$$ and also$$I\_{\infty}= \int \_0^{\infty}\exp(-X\_s)ds.$$When $X$ is a semi-martingale with absolutely continuous characteristics, we derive necessary and sufficient conditions for the existence of the Laplace exponent of $I\_t$, and also the sufficient conditions of finiteness of the Mellin transform ${\bf E}(I\_t^{\alpha})$ with $\alpha\in \mathbb{R}$. We give a recurrent integral equations for this Mellin transform. Then we apply these recurrent formulas to calculate the moments. We present also the corresponding results for the exponentials of Levy processes, which hold under less restrictive conditions then in \cite{BY}. In particular, we obtain an explicit formula for the moments of $I\_t$ and $I\_{\infty}$, and we precise the exact number of finite moments of $I\_{\infty}$.