IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v128y2018i1p306-331.html
   My bibliography  Save this article

On the refracted–reflected spectrally negative Lévy processes

Author

Listed:
  • Pérez, José-Luis
  • Yamazaki, Kazutoshi

Abstract

We study a combination of the refracted and reflected Lévy processes. Given a spectrally negative Lévy process and two boundaries, it is reflected at the lower boundary while, whenever it is above the upper boundary, a linear drift at a constant rate is subtracted from the increments of the process. Using the scale functions, we compute the resolvent measure, the Laplace transform of the occupation times as well as other fluctuation identities that will be useful in applied probability including insurance, queues, and inventory management.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:1:p:306-331
    DOI: 10.1016/j.spa.2017.03.024
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S030441491530079X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2017.03.024?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. Pistorius, M. R., 2003. "On doubly reflected completely asymmetric Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 131-143, September.
    2. 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.
    3. Bayraktar, Erhan & Kyprianou, Andreas E. & Yamazaki, Kazutoshi, 2014. "Optimal dividends in the dual model under transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 133-143.
    4. Florin Avram & Zbigniew Palmowski & Martijn R. Pistorius, 2007. "On the optimal dividend problem for a spectrally negative L\'{e}vy process," Papers math/0702893, arXiv.org.
    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. 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.
    2. 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.
    3. 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.

    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. 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.
    2. Baurdoux, Erik J. & Yamazaki, Kazutoshi, 2015. "Optimality of doubly reflected Lévy processes in singular control," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2727-2751.
    3. Kazutoshi Yamazaki, 2017. "Phase-type Approximation of the Gerber-Shiu Function," Papers 1701.02798, arXiv.org.
    4. 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.
    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. Avanzi, Benjamin & Lau, Hayden & Wong, Bernard, 2020. "Optimal periodic dividend strategies for spectrally positive Lévy risk processes with fixed transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 315-332.
    7. Wenyuan Wang & Yuebao Wang & Ping Chen & Xueyuan Wu, 2022. "Dividend and Capital Injection Optimization with Transaction Cost for Lévy Risk Processes," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 924-965, September.
    8. Ewa Marciniak & Zbigniew Palmowski, 2018. "On the Optimal Dividend Problem in the Dual Model with Surplus-Dependent Premiums," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 533-552, November.
    9. 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.
    10. 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.
    11. Bo, Lijun & Song, Renming & Tang, Dan & Wang, Yongjin & Yang, Xuewei, 2012. "Lévy risk model with two-sided jumps and a barrier dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 50(2), pages 280-291.
    12. Hernández, Camilo & Junca, Mauricio & Moreno-Franco, Harold, 2018. "A time of ruin constrained optimal dividend problem for spectrally one-sided Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 57-68.
    13. Kazutoshi Yamazaki, 2017. "Inventory Control for Spectrally Positive Lévy Demand Processes," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 212-237, January.
    14. Florin Avram & Dan Goreac & Juan Li & Xiaochi Wu, 2021. "Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model," Mathematics, MDPI, vol. 9(9), pages 1-27, April.
    15. Ewa Marciniak & Zbigniew Palmowski, 2016. "On the Optimal Dividend Problem in the Dual Model with Surplus-Dependent Premiums," Papers 1605.04584, arXiv.org.
    16. 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.
    17. Baurdoux, Erik J. & Yamazaki, Kazutoshi, 2015. "Optimality of doubly reflected Lévy processes in singular control," LSE Research Online Documents on Economics 61617, London School of Economics and Political Science, LSE Library.
    18. Benjamin Avanzi & Hayden Lau & Bernard Wong, 2020. "Optimal periodic dividend strategies for spectrally positive L\'evy risk processes with fixed transaction costs," Papers 2003.13275, arXiv.org, revised May 2020.
    19. Benjamin Avanzi & Hayden Lau & Bernard Wong, 2020. "Optimal periodic dividend strategies for spectrally negative L\'evy processes with fixed transaction costs," Papers 2004.01838, arXiv.org, revised Dec 2020.
    20. 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.

    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:128:y:2018:i:1:p:306-331. 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.