We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate depending on the loss function, whenever the backward equation only couples into the forward diffusion through the Y process. We generalize this result to fully-coupled drift coefficients, and give sufficient conditions for convergence under standard assumptions. The resulting conditions are directly verifiable for any equation. Consequently, unlike in earlier theory, our convergence analysis enables the treatment of FBSDEs stemming from stochastic optimal control problems. In particular, we provide a theoretical justification for the non-convergence of the deep BSDE method observed in recent literature, and present direct guidelines for when convergence can be guaranteed in practice. Our theoretical findings are supported by several numerical experiments in high-dimensional settings.

翻译：本文关注由Han等人(2018)提出的深度BSDE方法所逼近的高维耦合正倒向随机微分方程系统。Han和Long(2020)已证明，当倒向方程仅通过Y过程与前向扩散耦合时，深度BSDE方法诱导的误差可得到依赖于损失函数的后验估计。我们将此结果推广至全耦合漂移系数情形，并在标准假设下给出收敛的充分条件。所得条件对任意方程均可直接验证。因此，与先前理论不同，我们的收敛分析能够处理源自随机最优控制问题的FBSDE。特别地，我们为近期文献中观察到的深度BSDE方法不收敛现象提供了理论依据，并给出了实践中保证收敛性的直接指导原则。我们的理论发现得到了若干高维场景数值实验的支持。