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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:131

Article

Barker, A.¹  Savov, M.² 

[1] Univ Reading, Dept Math & Stat, Reading, Berks, England

[2] Bulgarian Acad Sci, Inst Math & Informat, Akad Georgi Bonchev Bl 8, Sofia 1113, Bulgaria

关键词: Levy processes;    Complex analysis;    Special functions;    Financial mathematics;   

DOI  :  10.1016/j.spa.2020.09.017

来源: Elsevier

【 摘 要 】

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 Levy processes. For a subordinator (a non-decreasing Levy process) (X-s)(s >= 0), we study its exponential functional, integral(t)(0)e(-Xs) ds, 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 Levy processes we believe that this work may turn out to be a step towards a more in-depth study of general exponential functionals of Levy processes on a finite time horizon. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free