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

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  • Barker, A.
  • Savov, M.

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.

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  • Barker, A. & Savov, M., 2021. "Bivariate Bernstein–gamma functions and moments of exponential functionals of subordinators," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 454-497.
    • Handle: RePEc:eee:spapps:v:131:y:2021:i:c:p:454-497
    DOI: 10.1016/j.spa.2020.09.017
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    References listed on IDEAS

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    1. Maulik, Krishanu & Zwart, Bert, 2006. "Tail asymptotics for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 156-177, February.
    2. D. Hackmann & A. Kuznetsov, 2014. "Asian options and meromorphic Lévy processes," Finance and Stochastics, Springer, vol. 18(4), pages 825-844, October.
    3. P. Salminen & L. Vostrikova, 2016. "On exponential functionals of processes with independent increments," Papers 1610.08732, arXiv.org, revised Mar 2018.
    4. Li, Zenghu & Xu, Wei, 2018. "Asymptotic results for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 108-131.
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    Cited by:

    1. Xu, Wei, 2023. "Asymptotics for exponential functionals of random walks," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 1-42.

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