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 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以通过正倒向随机微分方程（FBSDE）进行表述。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。本文针对该深度SMP-BSDE算法给出了收敛性结果，并将其性能与现有其他方法进行了比较。特别地，通过采用Han和Long（2020）中的策略，我们推导了后验估计，并证明总逼近误差可由损失泛函值与离散化误差共同界定。针对高维随机控制问题（包括漂移控制和扩散控制情形），我们给出了数值算例，结果表明该算法相较于现有方法具有更优越的性能。