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Deep PDE solution to BSDE

Author

Listed:
  • Maxim Bichuch

    (University at Buffalo)

  • Jiahao Hou

    (Johns Hopkins University)

Abstract

We numerically solve a high-dimensional backward stochastic differential equation (BSDE) by solving the corresponding partial differential equation (PDE) instead. To have a good approximation of the gradient of the solution of the PDE, we numerically solve a coupled PDE, consisting of the original semilinear parabolic PDE and the PDEs for its derivatives. We then prove the existence and uniqueness of the classical solution of this coupled PDE, and then show how to truncate the unbounded domain to a bounded one, so that the error between the original solution and that of the same coupled PDE but on the bounded domain, is small. We then solve this coupled PDE using neural networks, and proceed to establish a convergence of the numerical solution to the true solution. Finally, we test this on 100-dimensional Allen–Cahn equation, a nonlinear Black–Scholes equation and other examples. We also compare our results to the result of solving the BSDE directly.

Suggested Citation

  • Maxim Bichuch & Jiahao Hou, 2024. "Deep PDE solution to BSDE," Digital Finance, Springer, vol. 6(4), pages 727-758, December.
    • Handle: RePEc:spr:digfin:v:6:y:2024:i:4:d:10.1007_s42521-023-00098-6
    DOI: 10.1007/s42521-023-00098-6
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    References listed on IDEAS

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    1. 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.
    2. Christian Bender & Nikolaus Schweizer & Jia Zhuo, 2017. "A Primal–Dual Algorithm For Bsdes," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 866-901, July.
    3. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1793-1812, December.
    4. 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.
    5. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
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    JEL classification:

    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other

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