On exponential functionals of processes with independent increments

In this paper, we study the exponential functionals of the processes $X$ with independent increments, namely, $I_t= \int_0^t\exp\{-X_s\}\,ds,\ t\geq 0$, and also $I_{\infty}= \int_0^{\infty}\exp\{-X_s\}\,ds$. When $X$ is a semimartingale 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 $\mathbf{E}(I_t^{\alpha})$ with $\alpha\in \mathbf{R}$. We give recurrent integral equations for this Mellin transform. Then we apply these recurrent formulas to calculate the moments. We also present the corresponding results for the exponential functionals of Lévy processes, which hold under less restrictive conditions than in [J. Bertoin and M. Yor, Probab. Surv., 2 (2005), pp. 191–212]. In particular, we obtain an explicit formula for the moments of $I_t$ and $I_{\infty}$, and we give the precise number of finite moments of $I_{\infty}$.