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Asymptotic results for exponential functionals of Lévy processes

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  • Li, Zenghu
  • Xu, Wei

Abstract

The asymptotic behavior of expectations of some exponential functionals of a Lévy process is studied. The key point is the observation that the asymptotics only depend on the sample paths with slowly decreasing local infimum. We give not only the convergence rate but also the expression of the limiting coefficient. The latter is given in terms of some transformations of the Lévy process based on its renewal function. As an application, we give an exact evaluation of the decay rate of the survival probability of a continuous-state branching process in random environment with stable branching mechanism.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:1:p:108-131
    DOI: 10.1016/j.spa.2017.04.005
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    References listed on IDEAS

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    1. Fu, Zongfei & Li, Zenghu, 2010. "Stochastic equations of non-negative processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 306-330, March.
    2. Afanasyev, V.I. & Geiger, J. & Kersting, G. & Vatutin, V.A., 2005. "Functional limit theorems for strongly subcritical branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1658-1676, October.
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    Cited by:

    1. 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.
    2. Xu, Wei, 2023. "Asymptotics for exponential functionals of random walks," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 1-42.
    3. Natalia Cardona-Tobón & Juan Carlos Pardo, 2024. "Explosion Rates for Continuous-State Branching Processes in a Lévy Environment," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3395-3425, November.

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