IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v116y2006i12p1712-1742.html
   My bibliography  Save this article

Weak existence and uniqueness for forward-backward SDEs

Author

Listed:
  • Delarue, F.
  • Guatteri, G.

Abstract

We aim to establish the existence and uniqueness of weak solutions to a suitable class of non-degenerate deterministic FBSDEs with a one-dimensional backward component. The classical Lipschitz framework is partially weakened: the diffusion matrix and the final condition are assumed to be space Hölder continuous whereas the drift and the backward driver may be discontinuous in x. The growth of the backward driver is allowed to be at most quadratic with respect to the gradient term. The strategy holds in three different steps. We first build a well controlled solution to the associated PDE and as a by-product a weak solution to the forward-backward system. We then adapt the "decoupling strategy" introduced in the four-step scheme of Ma, Protter and Yong [J. Ma, P. Protter, J. Yong, Solving forward-backward stochastic differential equations explicitly -- a four step scheme, Probab. Theory Related Fields 98 (1994) 339-359] to prove uniqueness.

Suggested Citation

  • Delarue, F. & Guatteri, G., 2006. "Weak existence and uniqueness for forward-backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1712-1742, December.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:12:p:1712-1742
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(06)00066-4
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Delarue, François, 2002. "On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 209-286, June.
    2. Lejay, Antoine, 2004. "A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE," Stochastic Processes and their Applications, Elsevier, vol. 110(1), pages 145-176, March.
    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. Gobet, Emmanuel & Makhlouf, Azmi, 2010. "-time regularity of BSDEs with irregular terminal functions," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1105-1132, July.
    2. Bouchemella, Nadira & Raynaud de Fitte, Paul, 2014. "Weak solutions of backward stochastic differential equations with continuous generator," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 927-960.
    3. Luo, Peng & Menoukeu-Pamen, Olivier & Tangpi, Ludovic, 2022. "Strong solutions of forward–backward stochastic differential equations with measurable coefficients," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 1-22.
    4. Issoglio, Elena & Jing, Shuai, 2020. "Forward–backward SDEs with distributional coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 47-78.
    5. Martin Herdegen & Johannes Muhle-Karbe & Dylan Possamai, 2019. "Equilibrium Asset Pricing with Transaction Costs," Papers 1901.10989, arXiv.org, revised Sep 2020.

    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. Kupper, Michael & Luo, Peng & Tangpi, Ludovic, 2019. "Multidimensional Markovian FBSDEs with super-quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 902-923.
    2. Menozzi, Stéphane, 2018. "Martingale problems for some degenerate Kolmogorov equations," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 756-802.
    3. Umut c{C}etin & Albina Danilova, 2014. "Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems," Papers 1407.2420, arXiv.org, revised Sep 2016.
    4. Masaaki Fujii, 2020. "Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations," CARF F-Series CARF-F-497, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    5. Arvind V. Shrivats & Dena Firoozi & Sebastian Jaimungal, 2022. "A mean‐field game approach to equilibrium pricing in solar renewable energy certificate markets," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 779-824, July.
    6. Tomasz Klimsiak, 2013. "On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators," Journal of Theoretical Probability, Springer, vol. 26(2), pages 437-473, June.
    7. Masaaki Fujii, 2019. "Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations," CIRJE F-Series CIRJE-F-1133, CIRJE, Faculty of Economics, University of Tokyo.
    8. Ma, Jin & Yin, Hong & Zhang, Jianfeng, 2012. "On non-Markovian forward–backward SDEs and backward stochastic PDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 3980-4004.
    9. Delbaen, Freddy & Qiu, Jinniao & Tang, Shanjian, 2015. "Forward–backward stochastic differential systems associated to Navier–Stokes equations in the whole space," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2516-2561.
    10. Anne Eyraud-Loisel, 2013. "Quadratic hedging in an incomplete market derived by an influent informed investor," Post-Print hal-00450949, HAL.
    11. J. T. Shi & Z. Wu, 2010. "Maximum Principle for Partially-Observed Optimal Control of Fully-Coupled Forward-Backward Stochastic Systems," Journal of Optimization Theory and Applications, Springer, vol. 145(3), pages 543-578, June.
    12. Mikami, Toshio & Thieullen, Michèle, 2006. "Duality theorem for the stochastic optimal control problem," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1815-1835, December.
    13. Holger Kraft & Thomas Seiferling & Frank Thomas Seifried, 2017. "Optimal consumption and investment with Epstein–Zin recursive utility," Finance and Stochastics, Springer, vol. 21(1), pages 187-226, January.
    14. Rene Carmona & Francois Delarue & Gilles-Edouard Espinosa & Nizar Touzi, 2012. "Singular Forward-Backward Stochastic Differential Equations and Emissions Derivatives," Papers 1210.5773, arXiv.org.
    15. Delarue, François & Foguen Tchuendom, Rinel, 2020. "Selection of equilibria in a linear quadratic mean-field game," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 1000-1040.
    16. Horst, Ulrich & Hu, Ying & Imkeller, Peter & Réveillac, Anthony & Zhang, Jianing, 2014. "Forward–backward systems for expected utility maximization," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1813-1848.
    17. Yin, Hong, 2014. "Solvability of forward–backward stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2583-2604.
    18. Bruno Bouchard & Masaaki Fukasawa & Martin Herdegen & Johannes Muhle-Karbe, 2018. "Equilibrium Returns with Transaction Costs," Post-Print hal-01569408, HAL.
    19. Martin Herdegen & Johannes Muhle-Karbe & Dylan Possamai, 2019. "Equilibrium Asset Pricing with Transaction Costs," Papers 1901.10989, arXiv.org, revised Sep 2020.
    20. Alexander Fromm, 2019. "Evaluation of equity-based debt obligations," Papers 1901.02254, 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:spapps:v:116:y:2006:i:12:p:1712-1742. 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.