On distributions of exponential functionals of the processes with independent increments

The aim of this paper is to study the laws of exponential functionals of the processes X = (X-s)(s >= 0) with independent increments, namely I-t = integral(t)(0) exp(-X-s)ds, t >= 0, and also I-infinity = integral(infinity)(0) exp(-X-s)ds. Under suitable conditions, the integro-differential equations for the density of I-t and I-infinity are derived. Sufficient conditions are derived for the existence of a smooth density of the laws of these functionals with respect to the Lebesgue measure. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.