A Correction to "A Relative Value Iteration Algorithm for Nondegenerate Controlled Diffusions"

In A Relative Value Iteration Algorithm for Nondegenerate Controlled Diffusions, [SIAM J. Control Optim., 50 (2012), pp. 1886–1902], convergence of the relative value iteration for the ergodic control problem for a nondegenerate diffusion controlled through its drift was established, under the assumption of geometric ergodicity, using two methods: (a) the theory of monotone dynamical systems and (b) the theory of reverse martingales. However, in the proof using (a) it is wrongly claimed that the semiflow is strong order preserving. In this note, we provide a simple generic proof and also comment on how to relax the uniform geometric ergodicity hypothesis.