Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces

TL;DR

This paper introduces ergodic BSDEs in Banach spaces, explores their solutions, and applies them to optimal ergodic control problems, including stochastic PDEs, establishing links with HJB equations.

Contribution

It develops the theory of ergodic BSDEs in Banach spaces and applies it to solve optimal ergodic control problems for stochastic PDEs.

Findings

Existence and uniqueness of solutions to ergodic BSDEs established.

Connection between ergodic BSDEs and Hamilton-Jacobi-Bellman equations demonstrated.

Applications to ergodic control of stochastic PDEs provided.

Abstract

In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the ergodic BSDEs and the associated Hamilton-Jacobi-Bellman equation. Applications are given to ergodic control of stochastic partial differential equations.