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Implicit renewal theory for exponential functionals of Lévy processes

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  • Arista, Jonas
  • Rivero, Víctor

Abstract

We establish a new integral equation for the probability density of the exponential functional of a Lévy process and provide a three-term (Wiener–Hopf type) factorisation of its law. We explain how these results complement the techniques used in the study of exponential functionals and, in some cases, provide quick proofs of known results and derive new ones. We explain how the factors appearing in the three-term factorisation determine the local and asymptotic behaviour of the law of the exponential functional. We describe the behaviour of the tail distribution at infinity and of the distribution at zero under some mild assumptions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:163:y:2023:i:c:p:262-287
    DOI: 10.1016/j.spa.2023.06.004
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    References listed on IDEAS

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    3. Haas, Bénédicte & Rivero, Víctor, 2012. "Quasi-stationary distributions and Yaglom limits of self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 4054-4095.
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    5. Anita Behme & Alexander Lindner, 2015. "On Exponential Functionals of Lévy Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 681-720, June.
    6. Bertoin, J. & Doney, R. A., 1994. "Cramer's estimate for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 21(5), pages 363-365, December.
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