On moments of integral exponential functionals of additive processes

TL;DR

This paper derives recursive formulas for the moments of exponential functionals of additive processes, providing conditions for their finiteness and applying results to diffusion hit processes.

Contribution

It introduces a recursive equation for moments of exponential functionals of additive processes and offers a practical criterion for their finiteness, with applications to diffusion processes.

Findings

Recursive formula for moments of exponential functionals

Sufficient condition for moments' finiteness

Application to diffusion first hit processes

Abstract

For real-valued additive process $(X\_t)\_{t\geq 0}$ a recursive equation is derived for the entire positive moments of functionals $$I\_{s,t}= \int \_s^t\exp(-X\_u)du, \quad 0\leq s<t\leq\infty, $$ in case the Laplace exponent of $X\_t$ exists for positive values of the parameter. From the equation emergesan easy-to-apply sufficient condition for the finiteness of the moments. As an application we study first hitprocesses of diffusions.