On distributions of exponential functionals of the processes with independent increments

TL;DR

This paper investigates the probability distributions of exponential functionals of processes with independent increments, deriving equations for their densities and conditions for smoothness, with explicit solutions in the Levy process case.

Contribution

It introduces new integro-differential equations for the densities of exponential functionals and provides conditions for their smoothness, including explicit solutions for Levy processes.

Findings

Derived integro-differential equations for the densities.

Established conditions for the existence of smooth densities.

Provided explicit solutions in the Levy process case.

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.