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Tail asymptotics for exponential functionals of Lévy processes

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  • Maulik, Krishanu
  • Zwart, Bert

Abstract

Motivated by recent studies in financial mathematics and other areas, we investigate the exponential functional of a Lévy process X(t),t[greater-or-equal, slanted]0. In particular, we investigate its tail asymptotics. We show that, depending on the right tail of X(1), the tail behavior of Z is exponential, Pareto, or extremely heavy-tailed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:2:p:156-177
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    References listed on IDEAS

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    1. Veraverbeke, N., 1977. "Asymptotic behaviour of Wiener-Hopf factors of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 27-37, February.
    2. Willekens, Eric, 1987. "On the supremum of an infinitely divisible process," Stochastic Processes and their Applications, Elsevier, vol. 26, pages 173-175.
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    Cited by:

    1. Bertoin, Jean, 2019. "Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1443-1454.
    2. Kuznetsov, A., 2012. "On the distribution of exponential functionals for Lévy processes with jumps of rational transform," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 654-663.
    3. Tang, Qihe & Wang, Guojing & Yuen, Kam C., 2010. "Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 362-370, April.
    4. Krishanu Maulik & Bert Zwart, 2009. "An extension of the square root law of TCP," Annals of Operations Research, Springer, vol. 170(1), pages 217-232, September.
    5. Kyprianou, Andreas E. & Pardo, Juan Carlos, 2012. "An optimal stopping problem for fragmentation processes," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1210-1225.
    6. Lars Andersen, 2011. "Subexponential loss rate asymptotics for Lévy processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(1), pages 91-108, February.
    7. Chafaï, Djalil & Malrieu, Florent & Paroux, Katy, 2010. "On the long time behavior of the TCP window size process," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1518-1534, August.
    8. Feng, Runhuan & Kuznetsov, Alexey & Yang, Fenghao, 2019. "Exponential functionals of Lévy processes and variable annuity guaranteed benefits," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 604-625.
    9. Kamphorst, Bart & Zwart, Bert, 2019. "Uniform asymptotics for compound Poisson processes with regularly varying jumps and vanishing drift," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 572-603.
    10. Serguei Foss & Takis Konstantopoulos & Stan Zachary, 2007. "Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments," Journal of Theoretical Probability, Springer, vol. 20(3), pages 581-612, September.
    11. Fu, Ke-Ang & Ng, Cheuk Yin Andrew, 2014. "Asymptotics for the ruin probability of a time-dependent renewal risk model with geometric Lévy process investment returns and dominatedly-varying-tailed claims," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 80-87.
    12. Dyszewski, Piotr, 2016. "Iterated random functions and slowly varying tails," Stochastic Processes and their Applications, Elsevier, vol. 126(2), pages 392-413.
    13. 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.
    14. Palmowski, Zbigniew & Vlasiou, Maria, 2011. "A Lévy input model with additional state-dependent services," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1546-1564, July.
    15. 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.
    16. Runhuan Feng & Alexey Kuznetsov & Fenghao Yang, 2016. "Exponential functionals of Levy processes and variable annuity guaranteed benefits," Papers 1610.00577, arXiv.org.
    17. Li, Jinzhu, 2016. "Uniform asymptotics for a multi-dimensional time-dependent risk model with multivariate regularly varying claims and stochastic return," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 195-204.
    18. Anita Behme & Alexander Lindner, 2015. "On Exponential Functionals of Lévy Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 681-720, June.
    19. Abhishek Pal Majumder, 2024. "Long time behavior of semi-Markov modulated perpetuity and some related processes," Papers 2410.15824, arXiv.org, revised Feb 2025.

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