Moments of exponential functionals of Lévy processes on a deterministic horizon -- identities and explicit expressions (Q6589562)

The authors consider the moments of exponential functionals of Lévy processes on a deterministic time horizon. They derive two convolutional identities regarding these moments. The first one relates the complex moments of the exponential functional of a general Lévy process, up to a deterministic time, to those of the dual Lévy process; dual in the sense of negating the Lévy process. The second convolutional identity links the complex moments of the exponential functional of a Lévy process, which is not a compound Poisson process, to those of the exponential functionals of its ascending/descending ladder height process, on a random time horizon determined by the respective local times. Consequently, the authors derive a universal expression for the half-negative moment of the exponential functional of any symmetric Lévy process, which resembles the passage time of symmetric random walks. In addition, under extremely mild conditions, the authors obtain a series expansion for the complex moments (including those with negative real part) of the exponential functionals of subordinators. These results significantly extend previous results and offer neat expressions for the negative real moments.