Proper solutions for Epstein-Zin stochastic differential utility

Abstract

This article considers existence and uniqueness of infinite-horizon Epstein-Zin stochastic differential utility (EZ-SDU) for the case that the coefficients Rof relative risk aversion and Sof elasticity of intertemporal complementarity (the reciprocal of elasticity of intertemporal substitution) satisfy ϑ:=1-R1-S>1. In this sense, this paper is complementary to (Herdegen et al., Finance Stoch. 27, pp. 159-188). The main novelty of the case ϑ>1(as opposed to ϑ∈(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 proceed to show that for a very wide class of consumption streams C, there exists a proper utility process Vassociated to C. Furthermore, for a wide class of consumption streams C, the proper utility process Vis unique. Finally, we solve the optimal investment-consumption problem for an agent with preferences governed by EZ-SDU who invests in a constant-parameter Black-Scholes-Merton financial market and optimises over right-continuous consumption streams that have a unique proper utility process associated to them.