IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v17y2004i1d10.1023_bjotp.0000020481.14371.37.html
   My bibliography  Save this article

On Exit and Ergodicity of the Spectrally One-Sided Lévy Process Reflected at Its Infimum

Author

Listed:
  • M. R. Pistorius

    (King's College London, Strand)

Abstract

Consider a spectrally one-sided Lévy process X and reflect it at its past infimum I. Call this process Y. For spectrally positive X, Avram et al.(2) found an explicit expression for the law of the first time that Y=X−I crosses a finite positive level a. Here we determine the Laplace transform of this crossing time for Y, if X is spectrally negative. Subsequently, we find an expression for the resolvent measure for Y killed upon leaving [0,a]. We determine the exponential decay parameter ϱ for the transition probabilities of Y killed upon leaving [0,a], prove that this killed process is ϱ-positive and specify the ϱ-invariant function and measure. Restricting ourselves to the case where X has absolutely continuous transition probabilities, we also find the quasi-stationary distribution of this killed process. We construct then the process Y confined in [0,a] and prove some properties of this process.

Suggested Citation

  • M. R. Pistorius, 2004. "On Exit and Ergodicity of the Spectrally One-Sided Lévy Process Reflected at Its Infimum," Journal of Theoretical Probability, Springer, vol. 17(1), pages 183-220, January.
    • Handle: RePEc:spr:jotpro:v:17:y:2004:i:1:d:10.1023_b:jotp.0000020481.14371.37
    DOI: 10.1023/B:JOTP.0000020481.14371.37
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/B:JOTP.0000020481.14371.37
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/B:JOTP.0000020481.14371.37?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.

    Citations

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


    Cited by:

    1. Chongrui Zhu, 2022. "On the closed-form expected NPVs of double barrier strategies for regular diffusions," Papers 2206.08922, arXiv.org, revised Dec 2022.
    2. 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.
    3. Wenyuan Wang & Xiaowen Zhou, 2021. "A Drawdown Reflected Spectrally Negative Lévy Process," Journal of Theoretical Probability, Springer, vol. 34(1), pages 283-306, March.

    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:17:y:2004:i:1:d:10.1023_b:jotp.0000020481.14371.37. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.