Continuous-time reinforcement learning: ellipticity enables model-free value function approximation

We study off-policy reinforcement learning for controlling continuous-time Markov diffusion processes with discrete-time observations and actions. We consider model-free algorithms with function approximation that learn value and advantage functions directly from data, without unrealistic structural assumptions on the dynamics. Leveraging the ellipticity of the diffusions, we establish a new class of Hilbert-space positive definiteness and boundedness properties for the Bellman operators. Based on these properties, we propose the Sobolev-prox fitted $q$-learning algorithm, which learns value and advantage functions by iteratively solving least-squares regression problems. We derive oracle inequalities for the estimation error, governed by (i) the best approximation error of the function classes, (ii) their localized complexity, (iii) exponentially decaying optimization error, and (iv) numerical discretization error. These results identify ellipticity as a key structural property that renders reinforcement learning with function approximation for Markov diffusions no harder than supervised learning.