IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v150y2019icp137-145.html
   My bibliography  Save this article

On the first positive and negative excursion exceeding a given length

Author

Listed:
  • Sirovich, Roberta
  • Testa, Luisa

Abstract

For a one-dimensional diffusion process X, we derive the Laplace transform and the moments of the first time at which the age of an excursion above (or below) the level x is longer than u. The result is then illustrated for diffusion processes that are found relevant in applications. In the context of pricing Parisian options, the Brownian motion and the Geometric Brownian motion are considered and the Laplace transform can be made explicit and explicit expression for the moments can be derived. In the context of neuronal modeling, the Ornstein–Uhlenbeck process and the Cox–Ingersoll–Ross process are considered and the Laplace transform and the moments must be approximated by numerical inversion.

Suggested Citation

  • Sirovich, Roberta & Testa, Luisa, 2019. "On the first positive and negative excursion exceeding a given length," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 137-145.
    • Handle: RePEc:eee:stapro:v:150:y:2019:i:c:p:137-145
    DOI: 10.1016/j.spl.2019.03.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715219300847
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2019.03.008?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. Angelos Dassios & Jia Wei Lim, 2017. "An Analytical Solution For The Two-Sided Parisian Stopping Time, Its Asymptotics, And The Pricing Of Parisian Options," Mathematical Finance, Wiley Blackwell, vol. 27(2), pages 604-620, April.
    2. Angelos Dassios & Shanle Wu, 2010. "Perturbed Brownian motion and its application to Parisian option pricing," Finance and Stochastics, Springer, vol. 14(3), pages 473-494, September.
    3. Dassios, Angelos & Lim, Jia Wei, 2017. "An analytical solution for the two-sided Parisian stopping time, its asymptotics and the pricing of Parisian options," LSE Research Online Documents on Economics 60154, London School of Economics and Political Science, LSE Library.
    Full references (including those not matched with items on IDEAS)

    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. Gongqiu Zhang & Lingfei Li, 2021. "A General Approach for Parisian Stopping Times under Markov Processes," Papers 2107.06605, arXiv.org.
    2. Gongqiu Zhang & Lingfei Li, 2023. "A general approach for Parisian stopping times under Markov processes," Finance and Stochastics, Springer, vol. 27(3), pages 769-829, July.
    3. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.
    4. Angelos Dassios & Junyi Zhang, 2022. "First Hitting Time of Brownian Motion on Simple Graph with Skew Semiaxes," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1805-1831, September.
    5. Irmina Czarna & Zbigniew Palmowski, 2014. "Dividend Problem with Parisian Delay for a Spectrally Negative Lévy Risk Process," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 239-256, April.
    6. Angelos Dassios & Jia Wei Lim & Yan Qu, 2020. "Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero‐coupon bonds," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1497-1526, October.
    7. Anna Ananova & Rama Cont & Renyuan Xu, 2020. "Model-free Analysis of Dynamic Trading Strategies," Papers 2011.02870, arXiv.org, revised Aug 2023.
    8. Guglielmo D'Amico & Filippo Petroni, 2020. "A micro-to-macro approach to returns, volumes and waiting times," Papers 2007.06262, arXiv.org.
    9. Pingjin Deng & Xiufang Li, 2017. "Barrier Options Pricing With Joint Distribution Of Gaussian Process And Its Maximum," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(06), pages 1-18, September.
    10. Angelos Dassios & Jia Wei Lim, 2018. "An Efficient Algorithm for Simulating the Drawdown Stopping Time and the Running Maximum of a Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 189-204, March.
    11. Angelos Dassios & Luting Li, 2018. "Explicit Asymptotics on First Passage Times of Diffusion Processes," Papers 1806.08161, arXiv.org.
    12. Dassios, Angelos & Li, Luting, 2020. "Explicit asymptotic on first passage times of diffusion processes," LSE Research Online Documents on Economics 103087, London School of Economics and Political Science, LSE Library.
    13. B. A. Surya, 2018. "Parisian excursion below a fixed level from the last record maximum of Levy insurance risk process," Papers 1806.02083, arXiv.org.
    14. Angelos Dassios & Junyi Zhang, 2020. "Parisian Time of Reflected Brownian Motion with Drift on Rays and Its Application in Banking," Risks, MDPI, vol. 8(4), pages 1-14, December.
    15. Yangyang Zhuang & Pan Tang, 2023. "Pricing of American Parisian option as executive option based on the least‐squares Monte Carlo approach," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(10), pages 1469-1496, October.
    16. Czarna, Irmina & Palmowski, Zbigniew, 2017. "Parisian quasi-stationary distributions for asymmetric Lévy processes," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 75-84.
    17. Le, Nhat-Tan & Dang, Duy-Minh, 2017. "Pricing American-style Parisian down-and-out call options," Applied Mathematics and Computation, Elsevier, vol. 305(C), pages 330-347.
    18. Dassios, Angelos & Lim, Jia Wei & Qu, Yan, 2020. "Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero-coupon bonds," LSE Research Online Documents on Economics 101765, London School of Economics and Political Science, LSE Library.
    19. Angelos Dassios & Luting Li, 2018. "An Economic Bubble Model and Its First Passage Time," Papers 1803.08160, arXiv.org.

    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:stapro:v:150:y:2019:i:c:p:137-145. 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/622892/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.