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Hybrid Neural Networks for Solving Fully Coupled, High-Dimensional Forward–Backward Stochastic Differential Equations

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  • Mingcan Wang

    (School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430070, China)

  • Xiangjun Wang

    (School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430070, China)

Abstract

The theory of forward–backward stochastic differential equations occupies an important position in stochastic analysis and practical applications. However, the numerical solution of forward–backward stochastic differential equations, especially for high-dimensional cases, has stagnated. The development of deep learning provides ideas for its high-dimensional solution. In this paper, our focus lies on the fully coupled forward–backward stochastic differential equation. We design a neural network structure tailored to the characteristics of the equation and develop a hybrid BiGRU model for solving it. We introduce the time dimension based on the sequence nature after discretizing the FBSDE. By considering the interactions between preceding and succeeding time steps, we construct the BiGRU hybrid model. This enables us to effectively capture both long- and short-term dependencies, thus mitigating issues such as gradient vanishing and explosion. Residual learning is introduced within the neural network at each time step; the structure of the loss function is adjusted according to the properties of the equation. The model established above can effectively solve fully coupled forward–backward stochastic differential equations, effectively avoiding the effects of dimensional catastrophe, gradient vanishing, and gradient explosion problems, with higher accuracy, stronger stability, and stronger model interpretability.

Suggested Citation

  • Mingcan Wang & Xiangjun Wang, 2024. "Hybrid Neural Networks for Solving Fully Coupled, High-Dimensional Forward–Backward Stochastic Differential Equations," Mathematics, MDPI, vol. 12(7), pages 1-22, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1081-:d:1369679
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    References listed on IDEAS

    as
    1. 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.
    2. Hamadène, S., 1998. "Backward-forward SDE's and stochastic differential games," Stochastic Processes and their Applications, Elsevier, vol. 77(1), pages 1-15, September.
    3. 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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