SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 525-557, January 2021.
This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regression. This is achieved in the dynamic programming recursion by performance or hybrid iteration and regress-now methods from numerical probabilities. We provide a theoretical justification of these algorithms. Consistency and rate of convergence for the control and value function estimates are analyzed and expressed in terms of the universal approximation error of the neural networks, and of the statistical error when estimating network function, leaving aside the optimization error. Numerical results on various applications are presented in a companion paper [Deep neural networks algorithms for stochastic control problems on finite horizon: Numerical applications, Methodol. Comput. Appl. Probab., to appear] and illustrate the performance of the proposed algorithms.

中文翻译：

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有限时间范围内随机控制问题的深度神经网络算法：收敛性分析

SIAM数值分析学报，第59卷，第1期，第525-557页，2021年1月。
本文基于深度学习和动态规划，开发了用于高维随机控制问题的算法。与经典的近似动态规划方法不同，我们首先根据深度强化学习的精神借助神经网络对最佳策略进行近似，然后通过蒙特卡洛回归对价值函数进行近似。在动态编程递归中，这是通过性能或混合迭代和现值方法从数值概率中实现的。我们提供了这些算法的理论依据。对控制和值函数估计的一致性和收敛速度进行了分析，并根据神经网络的通用逼近误差以及估计网络函数时的统计误差（不包括优化误差）来表示。伴随论文中给出了各种应用的数值结果[有限域上的随机控制问题的深层神经网络算法：数值应用，Methodol。计算 应用 可能出现]，并说明所提出算法的性能。