On exponential functionals of processes with independent increments

TL;DR

This paper investigates exponential functionals of processes with independent increments, deriving integral equations for their Mellin transforms, calculating moments explicitly, and extending results to Levy processes under broader conditions.

Contribution

It introduces recurrent integral equations for Mellin transforms of exponential functionals of processes with independent increments, enabling explicit moment calculations and generalizing previous Levy process results.

Findings

Derived integral equations for Mellin transforms of exponential functionals.

Calculated explicit formulas for moments of $I_t$ and $I_{inite}$.

Identified the exact number of finite moments for $I_{inite}$.

Abstract

In this paper we study 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 $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ When $X$ is a semi-martingale with absolutely continuous characteristics, we derive recurrent integral equations for Mellin transform ${\bf E}( I_t^{\alpha})$, $\alpha\in\mathbb{R}$, of the integral functional $I_t$. Then we apply these recurrent formulas to calculate the moments. We present also the corresponding results for the exponential functionals of Levy processes, which hold under less restrictive conditions then in the paper of Bertoin, Yor (2005). In particular, we obtain an explicit formula for the moments of $I_t$ and $I_{\infty}$, and we precise the exact number of finite moments of $I_{\infty}$.