On distributions of exponential functionals of the processes with independent increments

The aim of this paper is to study the laws of 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 $$I_{\mathrm{\infty }}={\int }_0^{\mathrm{\infty }}\exp (-X_s)ds\mathrm{.}$$ Under suitable conditions we derive the integro-differential equations for the density of $I_t$ and $I_{\mathrm{\infty }}$. We give sufficient conditions for the existence of smooth density of the laws of these functionals. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.