On exponential functionals of processes with independent increments

In this paper we study the exponential functionals of the processes $X$ with independent increments , namely $$I_t={\int }_0^t\exp (-X_s)ds{,}_{,}t\ge 0,$$ and also $$I_{\mathrm{\infty }}={\int }_0^{\mathrm{\infty }}\exp (-X_s)ds\mathrm{.}$$ When $X$ is a semi-martingale with absolutely continuous characteristics, we derive recurrent integral equations for Mellin transform $\mathbf{E}(I_t^{\alpha })$, $\alpha \in \mathbb{R}$, of the integral functional $I_t$. Then we apply these recurrent formulas to calculate the moments. We present also the corresponding results for the exponential functionals of Levy processes, which hold under less restrictive conditions then in the paper of Bertoin, Yor (2005). In particular, we obtain an explicit formula for the moments of $I_t$ and $I_{\mathrm{\infty }}$, and we precise the exact number of finite moments of $I_{\mathrm{\infty }}$.