Optimal Comfortable Consumption under Epstein-Zin utility
Dejian Tian, Weidong Tian, Zimu Zhu
TL;DR
The paper advances the theory of optimal portfolio choice under Epstein-Zin utility with a dynamic minimum consumption constraint by developing a novel verification framework that combines HJB and dual approaches with a linearization method. It proves a $C^2$ smoothness property for the value function without an explicit solution and constructs a multi-layer candidate value function via a Fenchel-Legendre transform, reducing the HJB to two ODEs in the transformed space. A verification theorem establishes both sufficiency and necessity for the candidate, yielding a unique optimal strategy characterized by a free-boundary problem that separates constrained and unconstrained regions. The analysis reveals a two-region structure with a critical wealth boundary and a dual formulation for the constrained problem, and it demonstrates versatility for broader constrained portfolio problems where explicit solutions are unavailable.
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.