Ergodic BSDEs under weak dissipative assumptions

TL;DR

This paper extends the theory of ergodic backward stochastic differential equations by establishing existence and uniqueness results under weaker dissipativity conditions, with applications to stochastic control and PDEs.

Contribution

It introduces a novel approach to prove solutions for EBSDEs without requiring strong dissipativity, broadening their applicability.

Findings

Existence of solutions under non-degenerate conditions

Uniqueness of Markovian solutions via recurrence

Applications to ergodic control and Hamilton-Jacobi-Bellman equations

Abstract

In this paper we study ergodic backward stochastic differential equations (EBSDEs) dropping the strong dissipativity assumption needed in the previous work. In other words we do not need to require the uniform exponential decay of the difference of two solutions of the underlying forward equation, which, on the contrary, is assumed to be non degenerate. We show existence of solutions by use of coupling estimates for a non-degenerate forward stochastic differential equations with bounded measurable non-linearity. Moreover we prove uniqueness of "Markovian" solutions exploiting the recurrence of the same class of forward equations. Applications are then given to the optimal ergodic control of stochastic partial differential equations and to the associated ergodic Hamilton-Jacobi-Bellman equations.