On distributions of exponential functionals of the processes with independent increments

Abstract: The aim of this paper is to study the laws of the exponential functionals of the processes $X$ with independent increments, namely $$I_t= \int 0^t\exp(-X_s)ds, \,\, t\geq 0,$$ and also $$I{\infty}= \int 0^{\infty}\exp(-X_s)ds.$$ Under suitable conditions we derive the integro-differential equations for the density of $I_t$ and $I{\infty}$. 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.