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

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.

Keywords: Epstein--Zin stochastic differential utility, lifetime investment and consumption, backward stochastic differential equations, optional strong supermartingales.

JEL Classification: C61, G11