An Abelian Limit Approach to a Singular Ergodic Control Problem

We consider an ergodic stochastic control problem for a class of one-dimensional Itô processes where the available control is an added bounded variation process. The corresponding infinite horizon discounted control problem is solved in [28]. Here, we show that, as the discount factor approaches zero, the optimal strategies derived in [28] “converge” to an optimal strategy for the ergodic control problem. Under different assumptions, two types of optimal strategies were derived. Also, the Abelian limit relationships between the ergodic control problem, the infinite horizon discounted control problem and the finite time horizon control problem are established here. A solution to a constrained optimization problem is obtained as an application.