Optimal Comfortable Consumption under Epstein-Zin utility

TL;DR

This paper develops a new method for solving optimal portfolio problems with Epstein-Zin utility and time-varying consumption constraints, focusing on properties of the value function and verification without explicit solutions.

Contribution

It introduces a novel linearization approach to handle nonlinear HJB equations and characterizes the constrained region without explicit value function solutions.

Findings

Established $C^2$ smoothness of the value function

Proved verification theorem using linearization method

Characterized the constrained region explicitly

Abstract

We introduce a novel approach to solving the optimal portfolio choice problem under Epstein-Zin utility with a time-varying consumption constraint, where analytical expressions for the value function and the dual value function are not obtainable. We first establish several key properties of the value function, with a particular focus on the $C^2$ smoothness property. We then characterize the value function and prove the verification theorem by using the linearization method to the highly nonlinear HJB equation, despite the candidate value function being unknown a priori. Additionally, we present the sufficient and necessary conditions for the value function and explicitly characterize the constrained region. Our approach is versatile and can be applied to other portfolio choice problems with constraints where explicit solutions for both the primal and dual problems are unavailable.