Impulse Control of One-Dimensional Ito Diffusions with an Expected and a Pathwise Ergodic Criterion

We consider the problem of controlling a general one-dimensional Ito diffusion by means of an impulse control process. The objective is to minimise a long-term expected criterion as well as a long-term pathwise criterion that penalise both deviations of the state process from a given nominal point and the use of impulsive control effort. In particular, each time the controller deploys an impulse to reposition the system's state, a fixed cost and a cost proportional to the impulse's size are incurred. We solve the resulting optimisation problems and we provide an explicit characterisation of an optimal control strategy under general assumptions. The control of a foreign exchange rate or an inflation rate presents a potential application of the model that we study.