A reversible investment problem with general cost function in finite horizon: Free boundaries analysis

This paper focuses on optimizing reversible investments in a finite horizon, considering a quadratic cost function. The aim is to derive the optimal investment and divestment strategies for minimizing the overall expected cost. These strategies are characterized by two free boundaries, determined by solving a two-dimensional time-dependent Hamilton-Jacobi-Bellman (HJB) equation under gradient constraints. Due to spatial non-homogeneity, reducing the equation's dimension is infeasible. Employing the partial differential equation method, we establish the temporal continuity of the free boundaries, identify non-monotonicity cases, and determine their asymptotes. We also rigorously prove the strict monotonicity and continuity of the free boundaries with respect to spatial variables. The proposed approach can be extended to analyze situations with general cost functions.