  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.
