On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes

Abstract

We compute the positive entire moments of certain self-similar Markov processes evaluated at fixed time, and the negative entire moments of the exponential functional I of certain Lévy processes. When the Lévy process has no positive jumps, this determines the aforementioned distributions and yields several interesting identities in law. The case of the Poisson process yields yet another simple example showing that the log-normal distribution is moment-indeterminate.