A Solvable Two-Dimensional Degenerate Singular Stochastic Control Problem with Non Convex Costs

Abstract

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 and the smooth fit condition holds there.

Keywords: finite-fuel singular stochastic control, optimal stopping, free boundary, Hamilton-Jacobi-Bellmann equation, irreversible investment, electricity market