Proper solutions for Epstein-Zin Stochastic Differential Utility

TL;DR

This paper addresses the Epstein-Zin stochastic differential utility in continuous-time investment problems, introducing the concept of proper utility processes for the case where the risk aversion exceeds intertemporal elasticity, and establishing existence and uniqueness results.

Contribution

It introduces the notion of proper utility processes for Epstein-Zin utility with >1 and proves their existence and uniqueness for a broad class of consumption streams.

Findings

Existence of proper utility processes for a wide class of consumption streams.

Uniqueness of proper utility processes under certain conditions.

Solution to the optimal investment-consumption problem with Epstein-Zin utility for >1.

Abstract

In this article, we consider the optimal investment-consumption problem for an agent with preferences governed by Epstein--Zin stochastic differential utility (EZ-SDU) who invests in a constant-parameter Black-Scholes-Merton market over the infinite horizon. The parameter combinations that we consider in this paper are such that the risk aversion parameter $R$ and the elasticity of intertemporal complementarity $S$ satisfy $\theta=\frac{1-R}{1-S}>1$. In this sense, this paper is complementary to Herdegen, Hobson and Jerome [arXiv:2107.06593]. The main novelty of the case $\theta>1$ (as opposed to $\theta\in(0,1)$) is that there is an infinite family of utility processes associated to every nonzero consumption stream. To deal with this issue, we introduce the economically motivated notion of a proper utility process, where, roughly speaking, a utility process is proper if it is nonzero whenever future consumption is nonzero. We then proceed to show that for a very wide class of consumption streams $C$, there exists a proper utility process $V$ associated to $C$. Furthermore, for a wide class of consumption streams $C$, the proper utility process $V$ is unique. Finally, we solve the optimal investment-consumption problem in a constant parameter financial market, where we optimise over the right-continuous attainable consumption streams that have a unique proper utility process associated to them.