A Reversible Investment Problem with Capacity and Demand in Finite Horizon: Free Boundary Analysis

SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1207-1234, April 2024.
Abstract. This paper investigates a reversible investment problem with finite horizon, in which a social planner aims to determine the project’s capacity level to minimize the expected total costs. These costs depend on the demand for the good, the supply in terms of production capacity, and the proportional costs. The issue of irreversible investment has been examined by Han and Yi [Commun. Nonlinear Sci. Numer. Simul., 109 (2022), 106302]. Mathematically, the reversible investment problem can be formulated as a singular stochastic control problem. The value function satisfies a two-dimensional parabolic variational inequality subject to gradient constraint, which leads to two time-dependent free boundaries representing optimal investment and disinvestment strategies. We employ a partial differential equation approach to characterize the continuity, monotonicity, and horizontal asymptotes of free boundaries, as well as establish the [math] regularity of the value function. To the best of our knowledge, the approach to analyze the behavior of free boundaries is novel in the literature.