On Iteration Improvement for Averaged Expected Cost Control for One-Dimensional Ergodic Diffusions

An ergodic Bellman's (Hamilton--Jacobi--Bellman) equation is proved for a uniformly ergodic one-dimensional controlled diffusion with variable diffusion and drift coefficients both depending on control; convergence of the values provided by Howard's reward improvement algorithm to the value which is a component of the unique solution of Bellman's equation is established.