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Convergence of the deep BSDE method for stochastic control problems formulated through the stochastic maximum principle

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  • Huang, Zhipeng
  • Negyesi, Balint
  • Oosterlee, Cornelis W.

Abstract

It is well-known that decision-making problems from stochastic control can be formulated by means of a forward–backward stochastic differential equation (FBSDE). Recently, the authors of Ji et al. (2022) proposed an efficient deep learning algorithm based on the stochastic maximum principle (SMP). In this paper, we provide a convergence result for this deep SMP-BSDE algorithm and compare its performance with other existing methods. In particular, by adopting a strategy as in Han and Long (2020), we derive a-posteriori estimate, and show that the total approximation error can be bounded by the value of the loss functional and the discretization error. We present numerical examples for high-dimensional stochastic control problems, both in the cases of drift- and diffusion control, which showcase superior performance compared to existing algorithms.

Suggested Citation

  • Huang, Zhipeng & Negyesi, Balint & Oosterlee, Cornelis W., 2025. "Convergence of the deep BSDE method for stochastic control problems formulated through the stochastic maximum principle," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 227(C), pages 553-568.
    • Handle: RePEc:eee:matcom:v:227:y:2025:i:c:p:553-568
    DOI: 10.1016/j.matcom.2024.08.002
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    References listed on IDEAS

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    1. Maximilien Germain & Huyên Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance ," Post-Print hal-03115503, HAL.
    2. Maximilien Germain & Huy^en Pham & Xavier Warin, 2021. "Neural networks-based algorithms for stochastic control and PDEs in finance," Papers 2101.08068, arXiv.org, revised Apr 2021.
    3. Achref Bachouch & Côme Huré & Nicolas Langrené & Huyên Pham, 2022. "Deep Neural Networks Algorithms for Stochastic Control Problems on Finite Horizon: Numerical Applications," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 143-178, March.
    4. Bensoussan, A. & Yam, S.C.P. & Zhang, Z., 2015. "Well-posedness of mean-field type forward–backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3327-3354.
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