Stochastic internal habit formation and optimality

Growth models with internal habit formation have been studied in various settings under the assumption of deterministic dynamics. The purpose of this paper is to explore a stochastic version of the model in Carroll et al. [1997, 2000], one the most influential on the subject. The goal is twofold: on one hand, to determine how far we can advance in the technical study of the model; on the other, to assess whether at least some of the deterministic outcomes remain valid in the stochastic setting. The resulting optimal control problem proves to be challenging, primarily due to the lack of concavity in the objective function. This feature is present in the model even in the deterministic case (see, e.g., Bambi and Gozzi [2020]). We develop an approach based on Dynamic Programming to establish several useful results, including the regularity of the solution to the corresponding HJB equation and a verification theorem. There results lay the groundwork for studying the model optimal paths and comparing them with the deterministic case.