A singular control problem with an expected and a pathwise ergodicperformance criterion

We consider the problem of controlling a general one-dimensional Itô diffusion by means of a finite-variation process. The objective is to minimise a long-term average expected criterion as well as a long-term pathwise criterion that penalise deviations of the underlying state process from a given nominal point as well as the expenditure of control effort. We solve the resulting singular stochastic control problems under general assumptions by identifying an optimal strategy that is explicitly characterised.