A risk-sensitive ergodic singular stochastic control problem

Abstract

We consider a two-sided singular stochastic control problem with a risk-sensitive ergodic criterion. In particular, we consider a stochastic system whose uncontrolled dynamics are modelled by a linear diffusion. The control that can be applied to the system is modelled by an additive finite variation process. The objective of the control problem is to minimise a risk-sensitive long-term average criterion that penalises deviations of the controlled process from a given interval, as well as the expenditure of control effort. The stochastic control problem has been partly motivated by the problem faced by a central bank who wish to control the exchange rate between its domestic currency and a foreign currency so that this fluctuates within a suitable target zone. We derive the complete solution to the problem under general assumptions by deriving a C2 solution to its HJB equation. To this end, we use the solutions to a suitable family of Sturm-Liouville eigenvalue problems.