Proper solutions for Epstein–Zin stochastic differential utility

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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 ϑ : = 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 ϑ > 1 (as opposed to ϑ ∈ (0,1) 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 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.