The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. II: Existence, uniqueness and verification for ϑ ∈ ( 0 , 1 ) $\vartheta \in (0,1)$

In this article, we consider the optimal investment–consumption problem for an agent with preferences governed by Epstein–Zin (EZ) stochastic differential utility (SDU) over an infinite horizon. In a companion paper Herdegen et al. (Finance Stoch. 27:127–158, 2023), we argued that it is best to work with an aggregator in discounted form and that the coefficients R $R$ of relative risk aversion and S $S$ of elasticity of intertemporal complementarity (the reciprocal of the coefficient of elasticity of intertemporal substitution) must lie on the same side of unity for the problem to be well founded. This can be equivalently expressed as ϑ : = 1 − R 1 − S > 0 $\vartheta := \frac{1-R}{1-S} >0$ . In this paper, we focus on the case ϑ ∈ ( 0 , 1 ) $\vartheta \in (0,1)$ . The paper has three main contributions: first, to prove existence of infinite-horizon EZ SDU for a wide class of consumption streams and then (by generalising the definition of SDU) to extend this existence result to any consumption stream; second, to prove uniqueness of infinite-horizon EZ SDU for all consumption streams; and third, to verify the optimality of an explicit candidate solution to the investment–consumption problem in the setting of a Black–Scholes–Merton financial market.