Before revision:  For each policy , we show that a Partial Differential Equation (PDE) characterizes the evolution of the value function under that policy. Using the data , one can write down a sample version of the PDE that provides estimates of these value functions.
After revision:  For each policy , we show that a Partial Differential Equation (PDE) characterizes the evolution of the value function under each policy. The data enables us to obtain a sample version of the PDE that provides estimates of these value functions.