Ergodic BSDEs and related PDEs with Neumann boundary conditions

TL;DR

This paper introduces a new class of ergodic backward stochastic differential equations linked with Neumann boundary PDEs, establishing existence, uniqueness, and applications to ergodic control problems.

Contribution

It develops the theory of ergodic BSDEs with Neumann boundary conditions and connects them to PDEs, providing new tools for ergodic control.

Findings

Established existence and uniqueness of solutions to the new class of EBSDEs.

Demonstrated the link between EBSDEs and semi-linear PDEs with Neumann boundary conditions.

Applied the theoretical results to solve ergodic control problems.

Abstract

We study a new class of ergodic backward stochastic differential equations (EBSDEs for short) which is linked with semi-linear Neumann type boundary value problems related to ergodic phenomenas. The particularity of these problems is that the ergodic constant appears in Neumann boundary conditions. We study the existence and uniqueness of solutions to EBSDEs and the link with partial differential equations. Then we apply these results to optimal ergodic control problems.