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 modeled by a linear diffusion. The control that can be applied to the system is modeled by an additive finite variation process. The objective of the control problem is to minimize a risk-sensitive long-term average criterion that penalizes 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 that wishes 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 [Formula: see text] solution to its HJB equation. To this end, we use the solutions to a suitable family of Sturm–Liouville eigenvalue problems.