A simple proof of the theorem concerning optimality in a one-dimensional ergodic control problem (Q1968766)

The author considers the ergodic control problem to minimize the cost \[ J(v)= \limsup_{T\to \infty} E\int^T_0 f(x_t) dt, \] that refers to the one-dimensional stochastic differential equation \[ dx_t= b(x_t,\;v(x_t)) dt+ \sigma(x_t) dw_t,\;x_0= x, \] \[ v(\cdot)\in {\mathcal B}(\mathbb{R},\Gamma), \] where \(\Gamma\) is a compact set in a separable metric space. The function \(v\) is called a Markov control. The aim of the paper is to give a simple proof concerning optimality in the problem mentioned above. Although the class of controls is restricted to \({\mathcal B}(\mathbb{R},\Gamma)\), the proof is purely probabilistic and does not involve the solution of the Bellman equation. Let \[ b^*(y)= \xi\{y\geq 0\}\inf_{\gamma\in \Gamma} b(y,\gamma)+\chi\{y< 0\}\sup_{\gamma\in \Gamma} b(y, \gamma), \] and \(v^*(\cdot)\in {\mathcal B}(\mathbb{R}, \Gamma)\) be a function such that \(b^*(y)= b(y,v^*(y))\). Then it is established that \(J(v)\geq J(v^*)\), \(v\in{\mathcal B}(\mathbb{R}, \Gamma)\).