Before revision:  For each policy , we show that a Partial Differential Equation (PDE) characterizes the evolution of the value function under that policy.
After revision:  The problem is equivalent to one of optimal control under a constrained policy class, and we exploit recent developments in Reinforcement Learning (RL) to propose an algorithm to solve this. The algorithm is easily implementable and computationally efficient, with speedups achieved through multiple RL agents learning in parallel processes. We also characterize the statistical regret from using our estimated policy rule. To do this, we show that a Partial Differential Equation (PDE) characterizes the evolution of the value function under that policy.