On moments of exponential functionals of additive processes

Let X = (X t) t$\ge$0 be a real-valued additive process, i.e., a process with independent increments. In this paper we study the exponential integral functionals of X, namely, the functionals of the form I s,t = t s exp(--X u)du, 0 $\le$ s < t $\le$ $\infty$. Our main interest is focused on the moments of I s,t of order $\alpha$ $\ge$ 0. In the case when the Laplace exponent of X t is explicitly known, we derive a recursive (in $\alpha$) integral equation for the moments. This yields a multiple integral formula for the entire positive moments of I s,t. From these results emerges an easy-to-apply sufficient condition for the finiteness of all the entire moments of I $\infty$ := I 0,$\infty$. The corresponding formulas for L{\'e}vy processes are also presented. As examples we discuss the finiteness of the moments of I $\infty$ when X is the first hit process associated with a diffusion. In particular, we discuss the exponential functionals related with Bessel processes and geometric Brownian motions.