Learning to Solve Stochastic Controls with Unknown Drifts and Running Rewards: Theory, Algorithms and Convergence

2026-09-13

AI Summary

研究了未知漂移系数和运行奖励函数的随机控制问题，通过强化学习框架开发出有效的算法来学习最优值函数和控制策略。

Abstract

We study continuous-time and possibly high-dimensional stochastic control problems where drift coefficients and running reward functions are unknown. Due to these missing model primitives, we take the exploratory, reinforcement learning (RL) framework of Wang, Zariphopoulou, and Zhou(2020) with relaxed controls and entropy regularization. The objective is to develop theoretically grounded, efficient and scalable RL algorithms to learn both the optimal value functions (which also solve the exploratory HJB equation) and optimal exploratory feedback control policies. When the diffusion coefficients do not contain control, we employ probabilistic representations of both the optimal value function and its gradient based on an auxiliary state process depending only on the diffusion part of the original dynamics. With a delicate analysis on some properly defined mappings and their fixed points, this leads to the introduction of our policy iteration algorithms and their convergence. We demonstrate the performance of our algorithms through various numerical examples. Finally, we study a special control-dependent diffusion case where probability representation of the Hessian is called for.

Problem

• stochastic controls
• unknown drifts
• running rewards
• reinforcement learning
• optimal value functions

Innovation

• reinforcement learning
• entropy regularization
• exploratory HJB equation
• policy iteration algorithms
• probabilistic representation