Bivariate Bernstein–gamma functions and moments of exponential functionals of subordinators

A. Barker and M. Savov

Stochastic Processes and their Applications, 2021, vol. 131, issue C, 454-497

Abstract: In this paper, we extend recent work on the class of Bernstein–gamma functions to the class of bivariate Bernstein–gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise, improve upon, and streamline those found for univariate Bernstein–gamma functions. Then, we demonstrate the importance and power of these results through an application to exponential functionals of Lévy processes. For a subordinator (a non-decreasing Lévy process) (Xs)s≥0, we study its exponential functional, ∫0te−Xsds, evaluated at a finite, deterministic time t>0. Our main result here is an explicit infinite convolution formula for the Mellin transform (complex moments) of the exponential functional up to time t which is shown to be equivalent to an infinite series under very minor restrictions. Since exponential functionals of subordinators are part of a universal factorisation concerning all exponential functionals of Lévy processes we believe that this work may turn out to be a step towards a more in-depth study of general exponential functionals of Lévy processes on a finite time horizon.

Keywords: Lévy processes; Complex analysis; Special functions; Financial mathematics (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2020.09.017

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