Coupling and a generalised Policy Iteration Algorithm in continuous time

We analyse a version of the policy iteration algorithm for the discounted infinite-horizon problem for controlled multidimensional diffusion processes, where both the drift and the diffusion coefficient can be controlled. We prove that, under assumptions on the problem data, the payoffs generated by the algorithm converge monotonically to the value function and an accumulation point of the sequence of policies is an optimal policy. The algorithm is stated and analysed in continuous time and state, with discretisation featuring neither in theorems nor the proofs. A key technical tool used to show that the algorithm is well-defined is the mirror coupling of Lindvall and Rogers.