A New Monotonicity Condition for Ergodic Backward SDEs and Ergodic Control with Superquadratic Hamiltonians

Abstract

We establish the existence (and in an appropriate sense uniqueness) of Markovian solutions for ergodic backward SDEs (BSDEs) under a novel monotonicity condition. Our monotonicity condition allows us to prove existence even when the driver has arbitrary (in particular superquadratic) growth in , 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 superquadratic Hamiltonians. We also treat a class of drivers coming from the construction of forward performance processes and interpret our monotonicity condition in this setting.