In this paper, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs, fails for a simple linear-quadratic control problem, and motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

翻译：本文提出一种基于深度学习的数值格式，用于求解源自随机控制的强耦合前向-倒向随机微分方程。该方法是深度BSDE方法的改进版本，其中倒向方程的初值不作为自由参数，并引入由控制问题成本与终端条件均方误差项加权求和构成的新损失函数。通过数值算例表明，经典深度BSDE方法直接推广至FBSDE时会在简单线性二次控制问题上失效，并论证了新方法的有效性。在时间连续型及时间离散型控制问题的精确控制满足正则性和有界性假设下，我们给出了方法的误差分析。针对三个不同问题（包括导致深度BSDE方法直接推广失效的案例）的数值实验证明了该方法的收敛性。