Approximation of a stochastic ergodic control problem

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We study a degenerate non linear optimal stochastic control problem of ergodic type. We first prove that for each feedback control law, there exists a unique invariant measure which is equivalent to Lebesgue measure. This is proved using an accessibility property of the stochastic differential equation, after the discontinuous part of the drift has been removed via a change of probability measure. We then approximate the problem by ergodic control problems for finite state, continuous time Markov chains. We finally prove that the cost functionals of the approximate problems converge pointwise towards that of the continuous problem.