Topics in ergodic control and backward stochastic differential equations

The core of this thesis focuses on a number of different aspects of ergodic stochastic control in connection with backward stochastic differential equations (BSDEs for short). Chapter 1 serves as an introduction to the problem formulation in various contexts and states a number of results we will be using in the sequel. Chapter 2 deals with the so called weak formulation, where the control is represented as a change of measure. The optimal value and feedback control are obtained using a relatively recent object called ergodic BSDEs. In order to achieve this we establish the existence and uniqueness of solutions to these equations along the way. Chapter 3 is concerned with non zero-sum games where multiple players control the drift of a process, and their payoffs depend on its ergodic behaviour. We establish their connection with systems of Ergodic BSDEs, and prove the existence of a Nash equilibrium under general conditions. In Chapter 4 we show a novel duality between the existence of a solution to an infinite horizon adjoint BSDE and strong dissipativity of the forward process. Thus the link between ergodicity of the controlled process and the infinite horizon stochastic maximum principle is established. Finally, in Chapter 5 we provide conclusions, conjectures and directions for future research.