A Stochastic Reversible Investment Problem on a Finite-Time Horizon: Free Boundary Analysis

Abstract

We study a continuous-time, finite-horizon optimal stochastic reversible investment problem for a firm producing a single good. The production capacity is modeled as a one-dimensional, time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment-disinvestment strategy. We associate to the investment-disinvestment problem a zero-sum optimal stopping game and characterize its value function through a free boundary problem with two moving boundaries. These are continuous, bounded and monotone curves that solve a system of non-linear integral equations of Volterra type. The optimal investment-disinvestment strategy is then shown to be a diffusion reflected at the two boundaries.

Keywords: reversible investment, singular stochastic control, zero-sum optimal stopping games, free boundary problems, Skorokhod reflection problem

JEL Classification: C02, C73, E22, D92