Modern Stochastics: Theory and Applications

The aim of this paper is to study the laws of exponential functionals of the processes $X={({X_{s}})_{s\ge 0}}$ with independent increments, namely and also Under suitable conditions, the integro-differential equations for the density of ${I_{t}}$ and ${I_{\infty }}$ 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 Lévy processes these equations can be simplified and, in a number of cases, solved explicitly.