In this paper we study the exponential functionals of the processes X with independent increments , namelyI\_t = \int \_0 ^t\exp(-X \_s )ds, \_, \,\, t\geq 0, and alsoI\_ \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 \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 \mbox{%DIFAUXCMD \cite{BY}\hspace{0pt}%DIFAUXCMD }  . 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 .
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 recurrent integral equations for Mellin transform {\bf 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 }\hspace{0pt}%DIFAUXCMD } the paper of Bertoin, Yor (2005) . 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 .
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