Robust Exploratory Stopping under Ambiguity in Reinforcement Learning

We propose and analyze a continuous-time robust reinforcement learning framework for optimal stopping problems under ambiguity. In this framework, an agent chooses a stopping rule motivated by two objectives: robust decision-making under ambiguity and learning about the unknown environment. Here, ambiguity refers to considering multiple probability measures dominated by a reference measure, reflecting the agent's awareness that the reference measure representing her learned belief about the environment would be erroneous. Using the $g$-expectation framework, we reformulate an optimal stopping problem under ambiguity as an entropy-regularized optimal control problem under ambiguity, with Bernoulli distributed controls to incorporate exploration into the stopping rules. We then derive the optimal Bernoulli distributed control characterized by backward stochastic differential equations. Moreover, we establish a policy iteration theorem and implement it as a reinforcement learning algorithm. Numerical experiments demonstrate the convergence and robustness of the proposed algorithm across different levels of ambiguity and exploration.