  In this paper we provide a complete theoretical analysis of a two-dimensional
degenerate non convex singular stochastic control problem. The optimisation is
motivated by a storage-consumption model in an electricity market, and features
a stochastic real-valued spot price modelled by Brownian motion. We find
analytical expressions for the value function, the optimal control and the
boundaries of the action and inaction regions. The optimal policy is
characterised in terms of two monotone and discontinuous repelling free
boundaries, although part of one boundary is constant and the smooth fit
condition holds there.
