It is well-known that decision-making problems from stochastic control can be formulated by means of forward-backward stochastic differential equation (FBSDE). Recently, the authors of Ji et al. 2022 proposed an efficient deep learning-based algorithm which was 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 similar strategy as in Han and Long 2020, we derive a posteriori error 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 case of drift- and diffusion control, which showcase superior performance compared to existing algorithms.

翻译：众所周知，随机控制中的决策问题可通过前向-后向随机微分方程（FBSDE）进行建模。近期，Ji等人（2022）提出了一种基于随机最大值原理（SMP）的高效深度学习算法。本文给出了该深度SMP-BSDE算法的收敛性结果，并与其他现有方法进行了性能比较。具体而言，采用类似Han与Long（2020）的策略，我们推导了后验误差估计，证明总近似误差可由损失函数值与离散化误差共同界定。针对高维随机控制问题（包括漂移控制与扩散控制情形）的数值算例表明，该算法相较于现有方法具有更优性能。