The Finite-Horizon Reversible Investment Problem with the Constant Elasticity of Variance Model

Abstract. In this paper, we explore a two-dimensional parabolic Hamilton–Jacobi–Bellman (HJB) equation, constrained by two gradients, that arises from a firm’s finite-horizon reversible investment problem under economic uncertainty dynamics following the constant elasticity of variance (CEV) model or a similar diffusion model. The nature of the CEV model causes the differential operator of this HJB equation to exhibit degeneracy and singularity at zero. To tackle these challenges, we integrate a theory of the double obstacle problem with nonstandard arguments, examining the analytical properties of the HJB equation. Additionally, we show the smoothness of the two free boundaries associated with the HJB equation. Finally, we construct the solution to the original HJB equation, which corresponds to the solution of the firm’s problem, by utilizing the relationship between singular control and a family of switching control developed by [ 15 ], derived from the solution of a double obstacle problem.