A solvable two-dimensional singular stochastic control problem with non convex costs

TL;DR

This paper analyzes a complex two-dimensional stochastic control problem with non-convex costs, providing explicit solutions for the value function and optimal policy in a storage-consumption context with stochastic prices.

Contribution

It offers a complete theoretical solution including explicit formulas and boundary characterizations for a challenging non-convex control problem in a stochastic setting.

Findings

Explicit expressions for the value function and optimal control.

Characterization of free boundaries with monotonicity and discontinuities.

Identification of regions where control is active or inactive.

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 the smooth fit condition holds there.