IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v127y2017i2p643-656.html
   My bibliography  Save this article

Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations

Author

Listed:
  • Albrecher, Hansjörg
  • Ivanovs, Jevgenijs

Abstract

We consider exit problems for general Lévy processes, where the first passage over a threshold is detected either immediately or at an epoch of an independent homogeneous Poisson process. It is shown that the two corresponding one-sided problems are related through a surprisingly simple identity. Moreover, we identify a simple link between two-sided exit problems with one continuous and one Poisson exit. Finally, identities for reflected processes and a link between some Parisian type exit problems are established. For spectrally one-sided Lévy processes this approach enables alternative proofs for a number of previously established identities, providing additional insight.

Suggested Citation

  • Albrecher, Hansjörg & Ivanovs, Jevgenijs, 2017. "Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 643-656.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:2:p:643-656
    DOI: 10.1016/j.spa.2016.06.021
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915300235
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2016.06.021?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. David Landriault & Jean-François Renaud & Xiaowen Zhou, 2014. "An Insurance Risk Model with Parisian Implementation Delays," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 583-607, September.
    2. 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.
    3. Ivanovs, Jevgenijs & Mandjes, Michel, 2015. "Transient analysis of a stationary Lévy-driven queue," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 341-347.
    4. Albrecher, Hansjörg & Cheung, Eric C.K. & Thonhauser, Stefan, 2011. "Randomized Observation Periods for the Compound Poisson Risk Model: Dividends," ASTIN Bulletin, Cambridge University Press, vol. 41(2), pages 645-672, November.
    5. Ronnie Loeffen & Irmina Czarna & Zbigniew Palmowski, 2011. "Parisian ruin probability for spectrally negative L\'{e}vy processes," Papers 1102.4055, arXiv.org, revised Mar 2013.
    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. Lkabous, Mohamed Amine & Czarna, Irmina & Renaud, Jean-François, 2017. "Parisian ruin for a refracted Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 153-163.
    2. Mohamed Amine Lkabous, 2019. "Poissonian occupation times of spectrally negative L\'evy processes with applications," Papers 1907.09990, arXiv.org.
    3. Alvarez E., Luis H.R. & Lempa, Jukka & Saarinen, Harto & Sillanpää, Wiljami, 2024. "Solutions for Poissonian stopping problems of linear diffusions via extremal processes," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    4. Hamed Amini & Zhongyuan Cao & Andreea Minca & Agn`es Sulem, 2023. "Ruin Probabilities for Risk Processes in Stochastic Networks," Papers 2302.06668, arXiv.org.
    5. Zbigniew Palmowski & José Luis Pérez & Budhi Arta Surya & Kazutoshi Yamazaki, 2020. "The Leland–Toft optimal capital structure model under Poisson observations," Finance and Stochastics, Springer, vol. 24(4), pages 1035-1082, October.
    6. Eric C. K. Cheung & Jeff T. Y. Wong, 2023. "A Note on a Modified Parisian Ruin Concept," Risks, MDPI, vol. 11(3), pages 1-15, March.
    7. Nguyen, Giang T. & Peralta, Oscar, 2020. "An explicit solution to the Skorokhod embedding problem for double exponential increments," Statistics & Probability Letters, Elsevier, vol. 165(C).
    8. 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.
    9. 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.
    10. Hansjörg Albrecher & José Carlos Araujo-Acuna, 2022. "On The Randomized Schmitter Problem," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 515-535, June.
    11. Zbigniew Palmowski & Jos'e Luis P'erez & Budhi Arta Surya & Kazutoshi Yamazaki, 2019. "The Leland-Toft optimal capital structure model under Poisson observations," Papers 1904.03356, arXiv.org, revised Mar 2020.
    12. Boxma, Onno & Kella, Offer & Mandjes, Michel, 2023. "On fluctuation-theoretic decompositions via Lindley-type recursions," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 316-336.

    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. Guérin, Hélène & Renaud, Jean-François, 2017. "On the distribution of cumulative Parisian ruin," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 116-123.
    2. 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.
    3. 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.
    4. Xiaoqing Liang & Virginia R. Young, 2020. "Minimizing the Probability of Lifetime Exponential Parisian Ruin," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 1036-1064, March.
    5. Czarna, Irmina & Renaud, Jean-François, 2016. "A note on Parisian ruin with an ultimate bankruptcy level for Lévy insurance risk processes," Statistics & Probability Letters, Elsevier, vol. 113(C), pages 54-61.
    6. Mohamed Amine Lkabous & Irmina Czarna & Jean-Franc{c}ois Renaud, 2016. "Parisian ruin for a refracted L\'evy process," Papers 1603.09324, arXiv.org, revised Mar 2017.
    7. Li, Shu & Zhou, Xiaowen, 2022. "The Parisian and ultimate drawdowns of Lévy insurance models," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 140-160.
    8. Cheung, Eric C.K. & Zhu, Wei, 2023. "Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 84-101.
    9. Cheung, Eric C.K. & Wong, Jeff T.Y., 2017. "On the dual risk model with Parisian implementation delays in dividend payments," European Journal of Operational Research, Elsevier, vol. 257(1), pages 159-173.
    10. Brinker, Leonie Violetta & Eisenberg, Julia, 2021. "Dividend optimisation: A behaviouristic approach," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 202-224.
    11. Wong, Jeff T.Y. & Cheung, Eric C.K., 2015. "On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 280-290.
    12. 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.
    13. Lkabous, Mohamed Amine & Czarna, Irmina & Renaud, Jean-François, 2017. "Parisian ruin for a refracted Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 153-163.
    14. Landriault, David & Li, Bin & Shi, Tianxiang & Xu, Di, 2019. "On the distribution of classic and some exotic ruin times," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 38-45.
    15. Lkabous, Mohamed Amine & Wang, Zijia, 2023. "On the area in the red of Lévy risk processes and related quantities," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 257-278.
    16. Lkabous, Mohamed Amine, 2019. "A note on Parisian ruin under a hybrid observation scheme," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 147-157.
    17. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.
    18. Zhenyu Cui, 2014. "Omega risk model with tax," Papers 1403.7680, arXiv.org.
    19. Loeffen, R. & Palmowski, Z. & Surya, B.A., 2018. "Discounted penalty function at Parisian ruin for Lévy insurance risk process," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 190-197.
    20. Ran Xu & Wenyuan Wang & Jose Garrido, 2022. "Optimal Dividend Strategy Under Parisian Ruin with Affine Penalty," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1385-1409, September.

    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:127:y:2017:i:2:p:643-656. 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.