On moments of exponential functionals of additive processes
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Let X = (X t) t>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 < s < t < ∞. Our main interest is focused on the moments of I s,t of order α > 0. In the case when the Laplace exponent of X t is explicitly known, we derive a recursive (in α) 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 ∞ := I 0,∞. The corresponding formulas for Lévy processes are also presented. As examples we discuss the finiteness of the moments of I ∞ 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.
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