The Infinite Horizon Investment-Consumption Problem for Epstein-Zin Stochastic Differential Utility

TL;DR

This paper analyzes the infinite-horizon investment-consumption problem for agents with Epstein-Zin utility in a Black-Scholes market, establishing existence, uniqueness, and optimality of solutions under specific parameter restrictions.

Contribution

It introduces a new formulation of Epstein-Zin utility, proves existence and uniqueness under parameter restrictions, and verifies the optimal solution for the infinite-horizon problem.

Findings

Existence and uniqueness of Epstein-Zin utility under certain parameters

A new formulation of Epstein-Zin utility emphasizing parameter restrictions

Verification of the optimal investment-consumption strategy

Abstract

In this article we consider the optimal investment-consumption problem for an agent with preferences governed by Epstein-Zin stochastic differential utility who invests in a constant-parameter Black-Scholes-Merton market. The paper has three main goals: first, to provide a detailed introduction to infinite-horizon Epstein-Zin stochastic differential utility, including a discussion of which parameter combinations lead to a well-formulated problem; second, to prove existence and uniqueness of infinite horizon Epstein-Zin stochastic differential utility under a restriction on the parameters governing the agent's risk aversion and temporal variance aversion; and third, to provide a verification argument for the candidate optimal solution to the investment-consumption problem among all admissible consumption streams. To achieve these goals, we introduce a slightly different formulation of Epstein-Zin stochastic differential utility to that which is traditionally used in the literature. This formulation highlights the necessity and appropriateness of certain restrictions on the parameters governing the stochastic differential utility function.