IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v129y2019i12p5406-5449.html
   My bibliography  Save this article

Fluctuation theory for level-dependent Lévy risk processes

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414919301413
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2019.03.006?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Noba, Kei, 2021. "On the optimality of double barrier strategies for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 73-102.
    2. Jakubowski, Tomasz, 2007. "The estimates of the mean first exit time from a ball for the [alpha]-stable Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1540-1560, October.
    3. Kazutoshi Yamazaki, 2017. "Phase-type Approximation of the Gerber-Shiu Function," Papers 1701.02798, arXiv.org.
    4. Duhalde, Xan & Foucart, Clément & Ma, Chunhua, 2014. "On the hitting times of continuous-state branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4182-4201.
    5. Czarna, Irmina & Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "Optimality of multi-refraction control strategies in the dual model," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 148-160.
    6. Ma, Rugang, 2015. "Lamperti transformation for continuous-state branching processes with competition and applications," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 11-17.
    7. Bouasker, O. & Letifi, N. & Prigent, J.-L., 2016. "Optimal funding and hiring/firing policies with mean reverting demand," Economic Modelling, Elsevier, vol. 58(C), pages 569-579.
    8. Kei Noba & Jos'e-Luis P'erez & Xiang Yu, 2019. "On the bail-out dividend problem for spectrally negative Markov additive models," Papers 1901.03021, arXiv.org, revised Feb 2020.
    9. Zhuo Jin & Huafu Liao & Yue Yang & Xiang Yu, 2019. "Optimal Dividend Strategy for an Insurance Group with Contagious Default Risk," Papers 1909.09511, arXiv.org, revised Oct 2020.
    10. Bankovsky, Damien & Sly, Allan, 2009. "Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2544-2562, August.
    11. Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "On the refracted–reflected spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 306-331.
    12. José-Luis Pérez & Kazutoshi Yamazaki & Xiang Yu, 2018. "On the Bail-Out Optimal Dividend Problem," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 553-568, November.
    13. Jos'e-Luis P'erez & Kazutoshi Yamazaki & Xiang Yu, 2017. "On the Bail-Out Optimal Dividend Problem," Papers 1709.06348, arXiv.org, revised Jun 2018.
    14. Foucart, Clément & Vidmar, Matija, 2024. "Continuous-state branching processes with collisions: First passage times and duality," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    15. Avanzi, Benjamin & Pérez, José-Luis & Wong, Bernard & Yamazaki, Kazutoshi, 2017. "On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 148-162.
    16. Jacobsen, Martin & Jensen, Anders Tolver, 2007. "Exit times for a class of piecewise exponential Markov processes with two-sided jumps," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1330-1356, September.
    17. Avram, Florin & Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "Spectrally negative Lévy processes with Parisian reflection below and classical reflection above," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 255-290.
    18. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:129:y:2019:i:12:p:5406-5449. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.