IDEAS home Printed from https://ideas.repec.org/p/ehl/lserod/119468.html
   My bibliography  Save this paper

Lp optimal prediction of the last zero of a spectrally negative Lévy process

Author

Listed:
  • Baurdoux, Erik J.
  • Pedraza, José M.

Abstract

Given a spectrally negative Lévy process X drifting to infinity, (inspired on the early ideas of Shiryaev (2002)) we are interested in finding a stopping time that minimises the Lp distance (p > 1) with g, the last time X is negative. The solution is substantially more difficult compared to the case p = 1, for which it was shown by Baurdoux and Pedraza (2020) that it is optimal to stop as soon as X exceeds a constant barrier. In the case of p > 1 treated here, we prove that solving this optimal prediction problem is equivalent to solving an optimal stopping problem in terms of a two-dimensional strong Markov process that incorporates the length of the current positive excursion away from 0.We show that an optimal stopping time is now given by the first time that X exceeds a non-increasing and non-negative curve depending on the length of the current positive excursion away from 0. We further characterise the optimal boundary and the value function as the unique solution of a non-linear system of integral equations within a subclass of functions. As examples, the case of a Brownian motion with drift and a Brownian motion with drift perturbed by a Poisson process with exponential jumps are considered.

Suggested Citation

  • Baurdoux, Erik J. & Pedraza, José M., 2024. "Lp optimal prediction of the last zero of a spectrally negative Lévy process," LSE Research Online Documents on Economics 119468, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:119468
    as

    Download full text from publisher

    File URL: http://eprints.lse.ac.uk/119468/
    File Function: Open access version.
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Damien Lamberton & Mohammed Mikou, 2008. "The critical price for the American put in an exponential Lévy model," Finance and Stochastics, Springer, vol. 12(4), pages 561-581, October.
    2. Erhan Bayraktar & Hao Xing, 2009. "Regularity of the Optimal Stopping Problem for Jump Diffusions," Papers 0902.2479, arXiv.org, revised Mar 2012.
    3. Gerber, Hans U., 1990. "When does the surplus reach a given target?," Insurance: Mathematics and Economics, Elsevier, vol. 9(2-3), pages 115-119, September.
    4. Rama Cont & Ekaterina Voltchkova, 2005. "Integro-differential equations for option prices in exponential Lévy models," Finance and Stochastics, Springer, vol. 9(3), pages 299-325, July.
    5. Damien Lamberton & Mohammed Mikou, 2013. "Exercise boundary of the American put near maturity in an exponential Lévy model," Finance and Stochastics, Springer, vol. 17(2), pages 355-394, April.
    6. Kristoffer Glover & Hardy Hulley, 2014. "Optimal prediction of the last-passage time of a transient diffusion," Published Paper Series 2014-5, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    7. Masahiko Egami & Rusudan Kevkhishvili, 2020. "Time reversal and last passage time of diffusions with applications to credit risk management," Finance and Stochastics, Springer, vol. 24(3), pages 795-825, July.
    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. Baurdoux, Erik J. & Pedraza, José M., 2023. "Predicting the last zero before an exponential time of a spectrally negative Lévy process," LSE Research Online Documents on Economics 119290, London School of Economics and Political Science, LSE Library.
    2. Lukas Gonon & Christoph Schwab, 2021. "Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models," Finance and Stochastics, Springer, vol. 25(4), pages 615-657, October.
    3. Min Song & Rong Wu & Xin Zhang, 2008. "Total duration of negative surplus for the dual model," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 24(6), pages 591-600, November.
    4. Fernández Lexuri & Hieber Peter & Scherer Matthias, 2013. "Double-barrier first-passage times of jump-diffusion processes," Monte Carlo Methods and Applications, De Gruyter, vol. 19(2), pages 107-141, July.
    5. Damien Lamberton & Mohammed Mikou, 2013. "Exercise boundary of the American put near maturity in an exponential Lévy model," Finance and Stochastics, Springer, vol. 17(2), pages 355-394, April.
    6. Zakaria Marah, 2023. "American Exchange option driven by a L\'evy process," Papers 2307.10900, arXiv.org.
    7. Zailei Cheng, 2017. "Optimal dividends in the dual risk model under a stochastic interest rate," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-16, March.
    8. Kyriakos Georgiou & Athanasios N. Yannacopoulos, 2023. "Probability of Default modelling with L\'evy-driven Ornstein-Uhlenbeck processes and applications in credit risk under the IFRS 9," Papers 2309.12384, arXiv.org.
    9. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    10. 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.
    11. Peter K. Friz & Stefan Gerhold & Marc Yor, 2013. "How to make Dupire's local volatility work with jumps," Papers 1302.5548, arXiv.org.
    12. Dassios, Angelos & Wu, Shanle, 2008. "Parisian ruin with exponential claims," LSE Research Online Documents on Economics 32033, London School of Economics and Political Science, LSE Library.
    13. Denis Belomestny & Markus Reiß, 2006. "Spectral calibration of exponential Lévy models," Finance and Stochastics, Springer, vol. 10(4), pages 449-474, December.
    14. He, Jingmin & Wu, Rong & Zhang, Huayue, 2009. "Total duration of negative surplus for the risk model with debit interest," Statistics & Probability Letters, Elsevier, vol. 79(10), pages 1320-1326, May.
    15. Jose Cruz & Maria Grossinho & Daniel Sevcovic & Cyril Izuchukwu Udeani, 2022. "Linear and Nonlinear Partial Integro-Differential Equations arising from Finance," Papers 2207.11568, arXiv.org.
    16. Landriault, David & Shi, Tianxiang, 2015. "Occupation times in the MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 75-82.
    17. Hainaut, Donatien & Leonenko, Nikolai, 2020. "Option pricing in illiquid markets: a fractional jump-diffusion approach," LIDAM Discussion Papers ISBA 2020003, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    18. Walter Farkas & Ludovic Mathys & Nikola Vasiljević, 2021. "Intra‐Horizon expected shortfall and risk structure in models with jumps," Mathematical Finance, Wiley Blackwell, vol. 31(2), pages 772-823, April.
    19. Kathrin Glau, 2015. "Feynman-Kac formula for L\'evy processes with discontinuous killing rate," Papers 1502.07531, arXiv.org, revised Nov 2015.
    20. Nemat Safarov & Colin Atkinson, 2017. "Natural Gas-Fired Power Plants Valuation And Optimization Under Lévy Copulas And Regime Switching," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-38, February.

    More about this item

    Keywords

    Lévy processes; optimal prediction; optimal stopping; Support from the Department of Statistics of LSE and the LSE Ph.D. Studentship is gratefully acknowledged by José M. Pedraza.;
    All these keywords.

    JEL classification:

    • F3 - International Economics - - International Finance
    • G3 - Financial Economics - - Corporate Finance and Governance
    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

    Statistics

    Access and download statistics

    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:ehl:lserod:119468. 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: LSERO Manager (email available below). General contact details of provider: https://edirc.repec.org/data/lsepsuk.html .

    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.