IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v118y2008i1p53-75.html
   My bibliography  Save this article

Discrete-time approximation of decoupled Forward-Backward SDE with jumps

Author

Listed:
  • Bouchard, Bruno
  • Elie, Romuald

Abstract

We study a discrete-time approximation for solutions of systems of decoupled Forward-Backward Stochastic Differential Equations (FBSDEs) with jumps. Assuming that the coefficients are Lipschitz-continuous, we prove the convergence of the scheme when the number of time steps n goes to infinity. The rate of convergence is at least n-1/2+[epsilon], for any [epsilon]>0. When the jump coefficient of the first variation process of the forward component satisfies a non-degeneracy condition which ensures its inversibility, we achieve the optimal convergence rate n-1/2. The proof is based on a generalization of a remarkable result on the path-regularity of the solution of the backward equation derived by Zhang [J. Zhang, A numerical scheme for BSDEs, Annals of Applied Probability 14 (1) (2004) 459-488] in the no-jump case.

Suggested Citation

  • Bouchard, Bruno & Elie, Romuald, 2008. "Discrete-time approximation of decoupled Forward-Backward SDE with jumps," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 53-75, January.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:1:p:53-75
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(07)00052-X
    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. Anne Eyraud-Loisel, 2005. "Backward stochastic differential equations with enlarged filtration: Option hedging of an insider trader in a financial market with jumps," Post-Print hal-01298905, HAL.
    2. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    3. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    4. Buckdahn, R. & Pardoux, E., 1994. "BSDE's with jumps and associated integro-partial differential equations," SFB 373 Discussion Papers 1994,41, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    5. Eyraud-Loisel, Anne, 2005. "Backward stochastic differential equations with enlarged filtration: Option hedging of an insider trader in a financial market with jumps," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1745-1763, November.
    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. Lorenc Kapllani & Long Teng, 2020. "Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations," Papers 2010.01319, arXiv.org, revised Jun 2022.
    2. Dela Vega, Engel John C. & Elliott, Robert J., 2022. "Backward stochastic differential equations with regime-switching and sublinear expectations," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 278-298.
    3. Lorenc Kapllani & Long Teng, 2024. "A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations," Papers 2404.08456, arXiv.org.
    4. Madan, Dilip & Pistorius, Martijn & Stadje, Mitja, 2016. "Convergence of BSΔEs driven by random walks to BSDEs: The case of (in)finite activity jumps with general driver," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1553-1584.
    5. Bouchard Bruno & Tan Xiaolu & Zou Yiyi & Warin Xavier, 2017. "Numerical approximation of BSDEs using local polynomial drivers and branching processes," Monte Carlo Methods and Applications, De Gruyter, vol. 23(4), pages 241-263, December.
    6. Andrew Lesniewski & Anja Richter, 2016. "Managing counterparty credit risk via BSDEs," Papers 1608.03237, arXiv.org, revised Aug 2016.
    7. Stefan Geiss & Emmanuel Gobet, 2010. "Fractional smoothness and applications in finance," Papers 1004.3577, arXiv.org.
    8. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2019. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 26(3), pages 391-408, September.
    9. Bouchard, Bruno & Chassagneux, Jean-François, 2008. "Discrete-time approximation for continuously and discretely reflected BSDEs," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2269-2293, December.
    10. Jiang Wu & Fucheng Liao & Masayoshi Tomizuka, 2017. "Optimal preview control for a linear continuous-time stochastic control system in finite-time horizon," International Journal of Systems Science, Taylor & Francis Journals, vol. 48(1), pages 129-137, January.
    11. Polynice Oyono Ngou & Cody Hyndman, 2014. "A Fourier interpolation method for numerical solution of FBSDEs: Global convergence, stability, and higher order discretizations," Papers 1410.8595, arXiv.org, revised May 2022.
    12. Qiang Han & Shaolin Ji, 2022. "A Multi-Step Algorithm for BSDEs Based On a Predictor-Corrector Scheme and Least-Squares Monte Carlo," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2403-2426, December.
    13. Stefan Geiss & Emmanuel Gobet, 2011. "Fractional smoothness and applications in Finance," Post-Print hal-00474803, HAL.
    14. Jana Bielagk & Arnaud Lionnet & Gonçalo dos Reis, 2015. "Equilibrium pricing under relative performance concerns," Working Papers hal-01245812, HAL.
    15. Richter, Anja, 2014. "Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3578-3611.
    16. Monique Jeanblanc & Thibaut Mastrolia & Dylan Possamaï & Anthony Réveillac, 2015. "Utility Maximization With Random Horizon: A Bsde Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(07), pages 1-43, November.
    17. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs with Jumps," Papers 1510.03220, arXiv.org, revised Sep 2018.
    18. Ioannis Exarchos & Evangelos Theodorou & Panagiotis Tsiotras, 2019. "Stochastic Differential Games: A Sampling Approach via FBSDEs," Dynamic Games and Applications, Springer, vol. 9(2), pages 486-505, June.
    19. Anne Eyraud-Loisel, 2019. "How Does Asymmetric Information Create Market Incompleteness?," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 531-538, June.
    20. Andrew Lesniewski, 2020. "Epidemic control via stochastic optimal control," Papers 2004.06680, arXiv.org, revised May 2020.

    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:118:y:2008:i:1:p:53-75. 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.