On certain integral functionals of integer-valued subordinators

Abstract

It is known that the exponential functional of a Poisson process admits a probability density function in the form of an infinite series. In this paper, we obtain an explicit expression for the density function of the exponential functional of any integer-valued subordinator, and by extension, limit representations for that of an arbitrary pure-jump subordinator. With an added positive drift, the density function is expressed via piecewise basis functions governed by a functional relation. Closed-form density functions for these cases have been established only for a few special instances of L\'{e}vy processes in the past literature. Our work substantially advances this line of research by providing an analytical perspective on the distribution of a broad class of exponential L\'{e}vy functionals, also suggesting potential methodological extensions to general purely discontinuous L\'{e}vy processes. Moreover, we consider arbitrary decreasing functionals of integer-valued subordinators by deriving sufficient and necessary conditions for their convergence, which are then applied to obtain limit-series representations for the density functions of inverse-power functionals. The numerical performance of the proposed formulae is demonstrated through various examples of well-known distributions.