Proper solutions for Epstein–Zin stochastic differential utility
研究了当风险规避系数与跨期互补弹性系数满足特定条件时，Epstein-Zin随机微分效用的存在性和唯一性，并解决了最优投资消费问题。
Abstract 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 $\vartheta := \frac{1-R}{1-S}>1$ 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 $\vartheta >1$ ϑ > 1 (as opposed to $\vartheta \in (0,1)$ 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.