IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v30y2017i4d10.1007_s10959-016-0690-8.html
   My bibliography  Save this article

Occupation Times of General Lévy Processes

Author

Listed:
  • Lan Wu

    (Peking University)

  • Jiang Zhou

    (Peking University)

  • Shuang Yu

    (Peking University)

Abstract

For an arbitrary Lévy process X which is not a compound Poisson process, we are interested in its occupation times. We use a quite novel and useful approach to derive formulas for the Laplace transform of the joint distribution of X and its occupation times. Our formulas are compact, and more importantly, the forms of the formulas clearly demonstrate the essential quantities for the calculation of occupation times of X. It is believed that our results are important not only for the study of stochastic processes, but also for financial applications.

Suggested Citation

  • Lan Wu & Jiang Zhou & Shuang Yu, 2017. "Occupation Times of General Lévy Processes," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1565-1604, December.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:4:d:10.1007_s10959-016-0690-8
    DOI: 10.1007/s10959-016-0690-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-016-0690-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-016-0690-8?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. Li, Yingqiu & Zhou, Xiaowen & Zhu, Na, 2015. "Two-sided discounted potential measures for spectrally negative Lévy processes," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 67-76.
    2. Kuznetsov, A., 2012. "On the distribution of exponential functionals for Lévy processes with jumps of rational transform," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 654-663.
    3. Loeffen, Ronnie L. & Renaud, Jean-François & Zhou, Xiaowen, 2014. "Occupation times of intervals until first passage times for spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1408-1435.
    4. Landriault, David & Renaud, Jean-François & Zhou, Xiaowen, 2011. "Occupation times of spectrally negative Lévy processes with applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2629-2641, November.
    5. Ning Cai & Nan Chen & Xiangwei Wan, 2010. "Occupation Times of Jump-Diffusion Processes with Double Exponential Jumps and the Pricing of Options," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 412-437, May.
    6. L. Alili & A. E. Kyprianou, 2005. "Some remarks on first passage of Levy processes, the American put and pasting principles," Papers math/0508487, arXiv.org.
    7. Kuznetsov, A. & Peng, X., 2012. "On the Wiener–Hopf factorization for Lévy processes with bounded positive jumps," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2610-2638.
    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. Xie, Jiayi & Cui, Zhenyu & Zhang, Zhimin, 2022. "Some new infinite series expansions for the first passage time densities in a jump diffusion model with phase-type jumps," Applied Mathematics and Computation, Elsevier, vol. 429(C).

    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. David Landriault & Bin Li & Mohamed Amine Lkabous, 2019. "On occupation times in the red of L\'evy risk models," Papers 1903.03721, arXiv.org, revised Jul 2019.
    2. Landriault, David & Li, Bin & Lkabous, Mohamed Amine, 2020. "On occupation times in the red of Lévy risk models," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 17-26.
    3. Xuebing Kuang & Xiaowen Zhou, 2017. "n -Dimensional Laplace Transforms of Occupation Times for Spectrally Negative Lévy Processes," Risks, MDPI, vol. 5(1), pages 1-14, January.
    4. Li, Yingqiu & Zhou, Xiaowen & Zhu, Na, 2015. "Two-sided discounted potential measures for spectrally negative Lévy processes," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 67-76.
    5. Landriault, David & Shi, Tianxiang, 2015. "Occupation times in the MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 75-82.
    6. Mohamed Amine Lkabous, 2019. "Poissonian occupation times of spectrally negative L\'evy processes with applications," Papers 1907.09990, arXiv.org.
    7. Li, Yingqiu & Zhou, Xiaowen, 2014. "On pre-exit joint occupation times for spectrally negative Lévy processes," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 48-55.
    8. Li, Bo & Palmowski, Zbigniew, 2018. "Fluctuations of Omega-killed spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3273-3299.
    9. Li, Yingqiu & Wei, Yushao & Peng, Zhaohui, 2021. "Occupation times for spectrally negative Lévy processes on the last exit time," Statistics & Probability Letters, Elsevier, vol. 175(C).
    10. Neofytos Rodosthenous & Hongzhong Zhang, 2020. "When to sell an asset amid anxiety about drawdowns," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1422-1460, October.
    11. Cui, Zhenyu & Nguyen, Duy, 2016. "Omega diffusion risk model with surplus-dependent tax and capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 150-161.
    12. Guérin, Hélène & Renaud, Jean-François, 2017. "On the distribution of cumulative Parisian ruin," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 116-123.
    13. Zbigniew Palmowski & Jos'e Luis P'erez & Kazutoshi Yamazaki, 2020. "Double continuation regions for American options under Poisson exercise opportunities," Papers 2004.03330, arXiv.org.
    14. Yingchun Deng & Xuan Huang & Ya Huang & Xuyan Xiang & Jieming Zhou, 2020. "n-Dimensional Laplace Transforms of Occupation Times for Pre-Exit Diffusion Processes," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(1), pages 345-360, March.
    15. Chunhao Cai & Bo Li, 2018. "Occupation Times of Intervals Until Last Passage Times for Spectrally Negative Lévy Processes," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2194-2215, December.
    16. Long, Mingsi & Zhang, Hongzhong, 2019. "On the optimality of threshold type strategies in single and recursive optimal stopping under Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2821-2849.
    17. Landriault, David & Li, Bin & Lkabous, Mohamed Amine, 2021. "On the analysis of deep drawdowns for the Lévy insurance risk model," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 147-155.
    18. Neofytos Rodosthenous & Hongzhong Zhang, 2020. "When to sell an asset amid anxiety about drawdowns," Papers 2006.00282, arXiv.org.
    19. Landriault, David & Li, Bin & Wong, Jeff T.Y. & Xu, Di, 2018. "Poissonian potential measures for Lévy risk models," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 152-166.
    20. Landriault, David & Li, Bin & Lkabous, Mohamed Amine & Wang, Zijia, 2023. "Bridging the first and last passage times for Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 308-334.

    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:spr:jotpro:v:30:y:2017:i:4:d:10.1007_s10959-016-0690-8. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.