On moments of integral exponential functionals of additive processes

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.