IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-01403616.html
   My bibliography  Save this paper

Optimal stopping with f -expectations: the irregular case

Author

Listed:
  • Miryana Grigorova

    (Universität Bielefeld)

  • Peter Imkeller

    (Institut für Mathematik [Berlin] - TUB - Technical University of Berlin / Technische Universität Berlin)

  • Youssef Ouknine

    (Faculté des Sciences Semlalia [Marrakech] - UCA - Université Cadi Ayyad [Marrakech])

  • Marie-Claire Quenez

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité)

Abstract

We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $\xi$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $\xi$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $\xi$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American options with irregular pay-off in an imperfect market model.

Suggested Citation

  • Miryana Grigorova & Peter Imkeller & Youssef Ouknine & Marie-Claire Quenez, 2020. "Optimal stopping with f -expectations: the irregular case," Post-Print hal-01403616, HAL.
    • Handle: RePEc:hal:journl:hal-01403616
    DOI: 10.1016/j.spa.2019.05.001
    Note: View the original document on HAL open archive server: https://hal.science/hal-01403616v5
    as

    Download full text from publisher

    File URL: https://hal.science/hal-01403616v5/document
    Download Restriction: no
    File URL: https://libkey.io/10.1016/j.spa.2019.05.001?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
    ---><---

    References listed on IDEAS

    as
    1. Miryana Grigorova & Marie-Claire Quenez, 2017. "Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs," Papers 1705.03724, arXiv.org.
    2. Erhan Bayraktar & Song Yao, 2009. "Optimal Stopping for Non-linear Expectations," Papers 0905.3601, arXiv.org, revised Jan 2011.
    3. Miryana Grigorova & Marie-Claire Quenez, 2017. "Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs," Post-Print hal-01519215, HAL.
    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. Marzougue, Mohamed, 2023. "Non-linear Dynkin games over split stopping times," Statistics & Probability Letters, Elsevier, vol. 193(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. Grigorova, Miryana & Imkeller, Peter & Ouknine, Youssef & Quenez, Marie-Claire, 2018. "Optimal Stopping With ƒ-Expectations: the irregular case," Center for Mathematical Economics Working Papers 587, Center for Mathematical Economics, Bielefeld University.
    2. Grigorova, Miryana & Imkeller, Peter & Ouknine, Youssef & Quenez, Marie-Claire, 2020. "Optimal stopping with f-expectations: The irregular case," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1258-1288.
    3. Klebert Kentia & Christoph Kuhn, 2017. "Nash equilibria for game contingent claims with utility-based hedging," Papers 1707.09351, arXiv.org, revised Sep 2018.
    4. Bayraktar, Erhan & Yao, Song, 2015. "Doubly reflected BSDEs with integrable parameters and related Dynkin games," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4489-4542.
    5. Irina Penner & Anthony Reveillac, 2013. "Risk measures for processes and BSDEs," Papers 1304.4853, arXiv.org.
    6. Irina Penner & Anthony Réveillac, 2015. "Risk measures for processes and BSDEs," Finance and Stochastics, Springer, vol. 19(1), pages 23-66, January.
    7. Bayraktar, Erhan & Yao, Song, 2012. "Quadratic reflected BSDEs with unbounded obstacles," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1155-1203.
    8. Joffrey Derchu & Philippe Guillot & Thibaut Mastrolia & Mathieu Rosenbaum, 2020. "AHEAD : Ad-Hoc Electronic Auction Design," Papers 2010.02827, arXiv.org.
    9. Grigorova, Miryana & Imkeller, Peter & Quenez, Marie-Claire & Ouknine, Youssef, 2018. "Doubly Reflected BSDEs and $\mathcal{E}$$^ƒ$-Dynkin games: beyond the right-continuous case," Center for Mathematical Economics Working Papers 598, Center for Mathematical Economics, Bielefeld University.
    10. Ariel Neufeld & Mario Sikic, 2017. "Nonconcave Robust Optimization with Discrete Strategies under Knightian Uncertainty," Papers 1711.03875, arXiv.org, revised Apr 2019.
    11. Bayraktar, Erhan & Yao, Song, 2011. "Optimal stopping for non-linear expectations--Part I," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 185-211, February.
    12. Marcel Nutz & Jianfeng Zhang, 2012. "Optimal stopping under adverse nonlinear expectation and related games," Papers 1212.2140, arXiv.org, revised Sep 2015.
    13. Bayraktar, Erhan & Yao, Song, 2017. "Optimal stopping with random maturity under nonlinear expectations," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2586-2629.
    14. Denis Belomestny & Tobias Hübner & Volker Krätschmer, 2022. "Solving optimal stopping problems under model uncertainty via empirical dual optimisation," Finance and Stochastics, Springer, vol. 26(3), pages 461-503, July.
    15. Daniel Fernholz & Ioannis Karatzas, 2012. "Optimal arbitrage under model uncertainty," Papers 1202.2999, arXiv.org.
    16. Miryana Grigorova & Peter Imkeller & Youssef Ouknine & Marie-Claire Quenez, 2018. "Doubly Reflected BSDEs and ${\cal E}^{f}$-Dynkin games: beyond the right-continuous case," Working Papers hal-01497914, HAL.
    17. Denis Belomestny & Volker Kraetschmer, 2017. "Minimax theorems for American options in incomplete markets without time-consistency," Papers 1708.08904, arXiv.org.
    18. Miryana Grigorova & Peter Imkeller & Youssef Ouknine & Marie-Claire Quenez, 2016. "Optimal stopping with f -expectations: the irregular case," Papers 1611.09179, arXiv.org, revised Aug 2018.
    19. Quenez, Marie-Claire & Sulem, Agnès, 2014. "Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3031-3054.
    20. Tianyang Nie & Edward Kim & Marek Rutkowski, 2018. "Arbitrage-Free Pricing of Game Options in Nonlinear Markets," Papers 1807.05448, 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:hal:journl:hal-01403616. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.