Characterization of the optimal boundaries in reversible investment problems

TL;DR

This paper investigates a reversible investment control problem with general demand dynamics and costs, using viscosity solutions to characterize optimal boundaries, and provides explicit solutions in the quadratic cost case.

Contribution

It introduces a novel approach using viscosity solutions to analyze a complex degenerate stochastic control problem with explicit results for quadratic costs.

Findings

Explicit characterization of optimal boundaries in the quadratic cost case

Demonstration of the smooth-fit $C^2$ property for the value function

Structural insights into the free boundary in reversible investment problems

Abstract

This paper studies a {\it reversible} investment problem where a social planner aims to control its capacity production in order to fit optimally the random demand of a good. Our model allows for general diffusion dynamics on the demand as well as general cost functional. The resulting optimization problem leads to a degenerate two-dimensional bounded variation singular stochastic control problem, for which explicit solution is not available in general and the standard verification approach can not be applied a priori. We use a direct viscosity solutions approach for deriving some features of the optimal free boundary function, and for displaying the structure of the solution. In the quadratic cost case, we are able to prove a smooth-fit $C^2$ property, which gives rise to a full characterization of the optimal boundaries and value function.