Forward–backward doubly stochastic differential equations (FBDSDEs) serve as a probabilistic interpretation of stochastic partial differential equations (SPDEs) with diverse applications. Coupled FBDSDEs encounter numerous challenges in numerical approximation compared to forward–backward stochastic differential equations (FBSDEs) and decoupled FBDSDEs, including ensuring the measurability of the numerical solutions, accounting for the mutual influences between forward and backward processes, and considering the relationship with respect to SPDEs rather than PDEs. This paper introduces, for the first time, a numerical method for solving multi-dimensional coupled FBDSDEs. By integrating an optimal control-based approach with deep neural networks, it effectively addresses the coupling-related challenges between forward and backward equations. Computational examples of coupled FBDSDEs with explicit solutions demonstrate that the proposed deep learning-based numerical algorithm achieves commendable performance in terms of both accuracy and efficiency.

中文翻译：

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一种求解多维耦合前向-后向双 SDE 的深度学习方法


前向-后向双随机微分方程 (FBDSDE) 作为随机偏微分方程 (SPDE) 的概率解释，具有多种应用。与前向-后向随机微分方程（FBSDE）和解耦 FBDSDE 相比，耦合 FBDSDE 在数值逼近方面遇到了许多挑战，包括确保数值解的可测量性、考虑前向过程和后向过程之间的相互影响以及考虑与SPDE 而不是 PDE。本文首次介绍了求解多维耦合 FBDSDE 的数值方法。通过将基于最优控制的方法与深度神经网络相结合，它有效地解决了前向方程和后向方程之间的耦合相关挑战。具有显式解的耦合 FBDSDE 的计算示例表明，所提出的基于深度学习的数值算法在准确性和效率方面均取得了值得称赞的性能。