IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v161y2023icp424-450.html
   My bibliography  Save this article

Nonlinear BSDEs on a general filtration with drivers depending on the martingale part of the solution

Author

Listed:
  • Klimsiak, Tomasz
  • Rzymowski, Maurycy

Abstract

In the present paper, we consider multidimensional nonlinear backward stochastic differential equations (BSDEs) with a driver depending on the martingale part M of a solution. We assume that the nonlinear term is merely monotone continuous with respect to the state variable. As to the regularity of the driver with respect to the martingale variable, we consider a very general condition which permits path-dependence on “the future” of the process M as well as a dependence of its law (McKean–Vlasov-type equations). For such drivers, we prove the existence and uniqueness of a global solution (i.e. for any maturity T>0) to a BSDE with data satisfying natural integrability conditions.

Suggested Citation

  • Klimsiak, Tomasz & Rzymowski, Maurycy, 2023. "Nonlinear BSDEs on a general filtration with drivers depending on the martingale part of the solution," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 424-450.
    • Handle: RePEc:eee:spapps:v:161:y:2023:i:c:p:424-450
    DOI: 10.1016/j.spa.2023.04.011
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414923000790
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2023.04.011?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. Klimsiak, Tomasz, 2015. "Reflected BSDEs on filtered probability spaces," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4204-4241.
    2. Buckdahn, Rainer & Li, Juan & Peng, Shige, 2009. "Mean-field backward stochastic differential equations and related partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3133-3154, October.
    3. Bensoussan, Alain & Li, Yiqun & Yam, Sheung Chi Phillip, 2018. "Backward stochastic dynamics with a subdifferential operator and non-local parabolic variational inequalities," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 644-688.
    4. Briand, Ph. & Delyon, B. & Hu, Y. & Pardoux, E. & Stoica, L., 2003. "Lp solutions of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 109-129, November.
    5. Quenez, Marie-Claire & Sulem, Agnès, 2013. "BSDEs with jumps, optimization and applications to dynamic risk measures," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3328-3357.
    6. Royer, Manuela, 2006. "Backward stochastic differential equations with jumps and related non-linear expectations," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1358-1376, October.
    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. Safa Alsheyab & Tahir Choulli, 2021. "Reflected backward stochastic differential equations under stopping with an arbitrary random time," Papers 2107.11896, arXiv.org.
    2. Roxana Dumitrescu & Marie-Claire Quenez & Agnès Sulem, 2015. "Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 219-242, October.
    3. Kruse, T. & Popier, A., 2016. "Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2554-2592.
    4. Choukroun, Sébastien & Cosso, Andrea & Pham, Huyên, 2015. "Reflected BSDEs with nonpositive jumps, and controller-and-stopper games," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 597-633.
    5. Klimsiak, Tomasz, 2019. "Systems of quasi-variational inequalities related to the switching problem," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1259-1286.
    6. Roxana Dumitrescu & Marie-Claire Quenez & Agn`es Sulem, 2016. "BSDEs with default jump," Papers 1612.05681, arXiv.org, revised Sep 2017.
    7. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2016. "Existence and uniqueness results for BSDEs with jumps: the whole nine yards," Papers 1607.04214, arXiv.org, revised Nov 2018.
    8. Dirk Becherer & Martin Buttner & Klebert Kentia, 2016. "On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples," Papers 1607.06644, arXiv.org, revised Nov 2019.
    9. Yao, Song, 2017. "Lp solutions of backward stochastic differential equations with jumps," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3465-3511.
    10. 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.
    11. Lesedi Mabitsela & Calisto Guambe & Rodwell Kufakunesu, 2018. "A note on representation of BSDE-based dynamic risk measures and dynamic capital allocations," Papers 1808.04611, arXiv.org.
    12. Cohen, Samuel N., 2012. "Representing filtration consistent nonlinear expectations as g-expectations in general probability spaces," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1601-1626.
    13. Fujii, Masaaki & Takahashi, Akihiko, 2019. "Solving backward stochastic differential equations with quadratic-growth drivers by connecting the short-term expansions," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1492-1532.
    14. Monia Karouf, 2019. "Reflected and Doubly Reflected Backward Stochastic Differential Equations with Time-Delayed Generators," Journal of Theoretical Probability, Springer, vol. 32(1), pages 216-248, March.
    15. Fan, ShengJun, 2016. "Existence of solutions to one-dimensional BSDEs with semi-linear growth and general growth generators," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 7-15.
    16. Miryana Grigorova & Peter Imkeller & Elias Offen & Youssef Ouknine & Marie-Claire Quenez, 2015. "Reflected BSDEs when the obstacle is not right-continuous and optimal stopping," Papers 1504.06094, arXiv.org, revised May 2017.
    17. Buckdahn, Rainer & Chen, Yajie & Li, Juan, 2021. "Partial derivative with respect to the measure and its application to general controlled mean-field systems," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 265-307.
    18. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    19. Julia Eisenberg & Stefan Kremsner & Alexander Steinicke, 2021. "Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate," Papers 2108.00234, arXiv.org.
    20. Zixin Feng & Dejian Tian, 2021. "Optimal consumption and portfolio selection with Epstein-Zin utility under general constraints," Papers 2111.09032, arXiv.org, revised May 2023.

    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:161:y:2023:i:c:p:424-450. 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.