Proper solutions for Epstein–Zin stochastic differential utility

This article considers existence and uniqueness of infinite-horizon Epstein–Zin stochastic differential utility (EZ-SDU) for the case that the coefficients R $R$ of relative risk aversion and S $S$ of elasticity of intertemporal complementarity (the reciprocal of elasticity of intertemporal substitution) satisfy ϑ : = 1 − R 1 − S > 1 $\vartheta := \frac{1-R}{1-S}>1$ . 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 $\vartheta >1$ (as opposed to ϑ ∈ ( 0 , 1 ) $\vartheta \in (0,1)$ ) 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 proceed to show that for a very wide class of consumption streams C $C$ , there exists a proper utility process V $V$ associated to C $C$ . Furthermore, for a wide class of consumption streams C $C$ , the proper utility process V $V$ is 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.