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 $$I_t={\int }_0^t\exp (-X_s)ds,t\ge 0,$$ 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.

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