An irreversible investment problem with demand on a finite horizon: The optimal investment boundary analysis

This paper studies a continuous-time, finite horizon, irreversible investment problem where a social planner aims to minimize total expected costs of production capacity and demand. Our model allows for general diffusion dynamics on the demand as well as production capacity process controlled by a nondecreasing process representing the cumulative investment. Mathematically, it is a singular stochastic control problem whose value function satisfies a two-dimensional parabolic variational inequality with gradient constraint. The problem gives rise to a free boundary which stands for the optimal investment boundary. We use partial differential equation (PDE) approach to characterize some features of the free boundary and prove the C-2,C-1 regularity of the value function. To the best of our knowledge, the method to study the monotonicity of free boundary about the time direction is an innovation. (C) 2022 Elsevier B.V. All rights reserved.