A new monotonicity condition for ergodic BSDEs and ergodic control with super-quadratic Hamiltonians

TL;DR

This paper introduces a new monotonicity condition that ensures the existence and uniqueness of solutions for ergodic BSDEs with super-quadratic growth, enabling advances in ergodic control and forward performance modeling.

Contribution

The paper presents a novel monotonicity condition that guarantees solutions for ergodic BSDEs with super-quadratic drivers, expanding the scope of ergodic control applications.

Findings

Existence of Markovian solutions under the new condition

Application to ergodic control with super-quadratic Hamiltonians

Representation of derivatives for a priori estimates

Abstract

We establish the existence (and in an appropriate sense uniqueness) of Markovian solutions for ergodic BSDEs under a novel monotonicity condition. Our monotonicity condition allows us to prove existence even when the driver f has arbitrary (in particular super-quadratic) growth in z, which reveals an interesting trade-off between monotonicity and growth for ergodic BSDEs. The technique of proof is to establish a probabilistic representation of the derivative of the Markovian solution, and then use this representation to obtain a-priori estimates. Our study is motivated by applications to ergodic control, and we use our existence result to prove the existence of optimal controls for a class of ergodic control problems with potentially super-quadratic Hamiltonians. We also treat a class of drivers coming from the construction of forward performance processes, and interpret our monotonicity condition in this setting.