Convergence of the Relative Value Iteration for the Ergodic Control Problem of Nondegenerate Diffusions

We study the relative value iteration for the ergodic control problem under a nearmonotone running cost structure for a nondegenerate diffusion controlled through its drift. This algorithm takes the form of a quasi-linear parabolic Cauchy initial value problem in Rd. We show that this Cauchy problem stabilizes or, in other words, that the solution of the quasi-linear parabolic equation converges for every bounded initial condition in C2(Rd) to the solution of the Hamilton– Jacobi–Bellman equation associated with the ergodic control problem.