Moments of exponential functionals of L\'{e}vy processes on a deterministic horizon -- identities and explicit expressions

Abstract

In this work, we consider moments of exponential functionals of L\'{e}vy processes on a deterministic horizon. We derive two convolutional identities regarding these moments. The first one relates the complex moments of the exponential functional of a general L\'{e}vy process up to a deterministic time to those of the dual L\'{e}vy process. The second convolutional identity links the complex moments of the exponential functional of a L\'{e}vy process, which is not a compound Poisson process, to those of the exponential functionals of its ascending/descending ladder heights on a random horizon determined by the respective local times. As a consequence, we derive a universal expression for the half-negative moment of the exponential functional of any symmetric L\'{e}vy process, which resembles in its universality the passage time of symmetric random walks. The $(n-1/2)^{th}$, $n\geq 0$ moments are also discussed. On the other hand, under extremely mild conditions, we obtain a series expansion for the complex moments (including those with negative real part) of the exponential functionals of subordinators. This significantly extends previous results and offers neat expressions for the negative real moments. In a special case, it turns out that the Riemann zeta function is the minus first moment of the exponential functional of the Gamma subordinator indexed in time.