Decoupled Continuous-Time Reinforcement Learning via Hamiltonian Flow

Many real-world control problems, ranging from finance to robotics, evolve in continuous time with non-uniform, event-driven decisions. Standard discrete-time reinforcement learning (RL), based on fixed-step Bellman updates, struggles in this setting: as time gaps shrink, the Q-function collapses to the value function V, eliminating action ranking. Existing continuous-time methods reintroduce action information via an advantage-rate function q. However, they enforce optimality through complicated martingale losses or orthogonality constraints, which are sensitive to the choice of test processes. These approaches entangle V and q into a large, complex optimization problem that is difficult to train reliably. To address these limitations, we propose a novel decoupled continuous-time actor-critic algorithm with alternating updates: q is learned from diffusion generators on V, and V is updated via a Hamiltonian-based value flow that remains informative under infinitesimal time steps, where standard max/softmax backups fail. Theoretically, we prove rigorous convergence via new probabilistic arguments, sidestepping the challenge that generator-based Hamiltonians lack Bellman-style contraction under the sup-norm. Empirically, our method outperforms prior continuous-time and leading discrete-time baselines across challenging continuous-control benchmarks and a real-world trading task, achieving 21% profit over a single quarter-nearly doubling the second-best method.