Convergence of The Relative Value Iteration for the Ergodic Control Problem of Nondegenerate Diffusions under Near-Monotone Costs

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 quasilinear parabolic Cauchy initial value problem in R. We show that this Cauchy problem stabilizes, or in other words, that the solution of the quasilinear parabolic equation converges for every bounded initial condition in C(R) to the solution of the Hamilton– Jacobi–Bellman (HJB) equation associated with the ergodic control problem.