# Yet again on iteration improvement for averaged expected cost control for 1D ergodic diffusions

## TL;DR

This paper advances ergodic control for one-dimensional diffusions by establishing the ergodic HJB equation, proving solution existence and uniqueness, and demonstrating convergence of a reward improvement algorithm.

## Contribution

It provides a comprehensive treatment of ergodic control with strategy-dependent coefficients, including theoretical foundations and convergence results.

## Key findings

- Proved existence and uniqueness of solutions to the ergodic HJB equation.
- Established convergence of the reward improvement algorithm.
- Extended ergodic control theory to strategy-dependent drift and diffusion coefficients.

## Abstract

The paper is a full version of the short presentation in \cite{amv17}. Ergodic control for one-dimensional controlled diffusion is tackled; both drift and diffusion coefficients may depend on a strategy which is assumed markovian. Ergodic HJB equation is established and existence and uniqueness of its solution is proved, as well as the convergence of the reward improvement algorithm.