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On Exponential Functionals of Lévy Processes

Author

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  • Anita Behme

    (Technische Universität München)

  • Alexander Lindner

    (Technische Universität Braunschweig)

Abstract

Exponential functionals of Lévy processes appear as stationary distributions of generalized Ornstein–Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further, we use these results to investigate properties of the mapping $$\Phi $$ Φ , which maps two independent Lévy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective, and give the inverse mapping in terms of (Lévy) characteristics. Also, continuity of $$\Phi $$ Φ is treated, and some results on its range are obtained.

Suggested Citation

  • Anita Behme & Alexander Lindner, 2015. "On Exponential Functionals of Lévy Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 681-720, June.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:2:d:10.1007_s10959-013-0507-y
    DOI: 10.1007/s10959-013-0507-y
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    References listed on IDEAS

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    3. Lindner, Alexander & Maller, Ross, 2005. "Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1701-1722, October.
    4. Medvegyev, Peter, 2007. "Stochastic Integration Theory," OUP Catalogue, Oxford University Press, number 9780199215256, Decembrie.
    5. Sato, Ken-iti & Yamazato, Makoto, 1984. "Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 73-100, May.
    6. Gerold Alsmeyer & Alex Iksanov & Uwe Rösler, 2009. "On Distributional Properties of Perpetuities," Journal of Theoretical Probability, Springer, vol. 22(3), pages 666-682, September.
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    Cited by:

    1. Arista, Jonas & Rivero, Víctor, 2023. "Implicit renewal theory for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 262-287.
    2. Yang Yang & Shuang Liu & Kam Chuen Yuen, 2022. "Second-Order Tail Behavior for Stochastic Discounted Value of Aggregate Net Losses in a Discrete-Time Risk Model," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2600-2621, December.

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