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Fluctuation theory for level-dependent Lévy risk processes

Author

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  • Czarna, Irmina
  • Pérez, José-Luis
  • Rolski, Tomasz
  • Yamazaki, Kazutoshi

Abstract

A level-dependent Lévy process solves the stochastic differential equation dU(t)=dX(t)−ϕ(U(t))dt, where X is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with ϕk(x)=∑j=1kδj1{x≥bj}. A general rate function ϕ that is non-decreasing and locally Lipschitz continuous is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of Lévy processes. We show how fluctuation identities for U can be expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations.

Suggested Citation

  • Czarna, Irmina & Pérez, José-Luis & Rolski, Tomasz & Yamazaki, Kazutoshi, 2019. "Fluctuation theory for level-dependent Lévy risk processes," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5406-5449.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:12:p:5406-5449
    DOI: 10.1016/j.spa.2019.03.006
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    References listed on IDEAS

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    1. Pérez, José-Luis & Yamazaki, Kazutoshi, 2017. "Refraction–Reflection Strategies In The Dual Model," ASTIN Bulletin, Cambridge University Press, vol. 47(1), pages 199-238, January.
    2. Patie, Pierre, 2005. "On a martingale associated to generalized Ornstein-Uhlenbeck processes and an application to finance," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 593-607, April.
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    Cited by:

    1. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    2. Irmina Czarna & Adam Kaszubowski, 2020. "Optimality of Impulse Control Problem in Refracted Lévy Model with Parisian Ruin and Transaction Costs," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 982-1007, June.
    3. Noba, Kei, 2023. "On the optimality of the refraction–reflection strategies for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 174-217.

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