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Stochastic internal habit formation and optimality

Michele Aleandri, Alessandro Bondi, Fausto Gozzi

TL;DR

The paper extends the Carroll endogenous growth model with internal habit formation to a stochastic setting by introducing multiplicative noise in the state dynamics of capital and habit. Using dynamic programming, it formulates a stochastic optimal control problem with a nonconcave objective and establishes a rigorous regularity theory for the value function $V$: first as a viscosity solution to the HJB equation in the first quadrant, then as a classical $C^2$ solution on $\mathbb{R}^2_{++}$ via local regularization (Safonov and Crandall techniques). A verification theorem provides a sufficient condition for optimality and connects the stochastic problem to closed-loop control via a feedback map $\Phi_v$. The results offer a solid foundation for analyzing model-implied optimal paths and for comparing stochastic outcomes with the known deterministic results, with future work aimed at deeper pathwise analysis and deterministic-stochastic comparisons.

Abstract

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.

Stochastic internal habit formation and optimality

TL;DR

The paper extends the Carroll endogenous growth model with internal habit formation to a stochastic setting by introducing multiplicative noise in the state dynamics of capital and habit. Using dynamic programming, it formulates a stochastic optimal control problem with a nonconcave objective and establishes a rigorous regularity theory for the value function : first as a viscosity solution to the HJB equation in the first quadrant, then as a classical solution on via local regularization (Safonov and Crandall techniques). A verification theorem provides a sufficient condition for optimality and connects the stochastic problem to closed-loop control via a feedback map . The results offer a solid foundation for analyzing model-implied optimal paths and for comparing stochastic outcomes with the known deterministic results, with future work aimed at deeper pathwise analysis and deterministic-stochastic comparisons.

Abstract

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.

Paper Structure

This paper contains 10 sections, 13 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. The model
  3. HJB equation: Formal derivation
  4. Properties of the Value function $V$
  5. Solving the HJB equation
  6. $V$ is a viscosity solution of the HJB equation
  7. $V$ is a classical solution of the HJB equation
  8. Verification theorem
  9. Conclusions
  10. Proof of Lemmas \ref{['lemm:safonov']} and \ref{['lemm:userguide']}

Key Result

Lemma 2.1

For every $(k_0,\,h_0)\in \mathbb{R}^2_{++}$, the set $\mathcal{A}(k_0,h_0)$ of admissible controls is nonempty. In addition, $\mathcal{A}(k_0,h_0)$ does not depend on $h_0$, that is,

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 16 more