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<p>Recent studies highlight the difficulties in infinite-horizon Epstein–Zin stochastic differential utility from economic and mathematical perspectives when the coefficient of relative risk aversion and the elasticity of intertemporal substitution both exceed&#xa0;1 (i.e., <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>θ</mi> <mo>&lt;</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\theta &lt; 0$</EquationSource> </InlineEquation>). We demonstrate that the economically problematic behaviour identified in recent studies disappears by an order-equivalent transformation of the utility index. Furthermore, we introduce an admissible set of consumptions to tackle mathematical issues for the existence of the utility. For applications, we examine a Merton problem and demonstrate that an optimal control derived from the Hamilton–Jacobi–Bellman equation is admissible and optimal under mild conditions.</p>

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An economic interpretation and mathematical analysis of Epstein–Zin stochastic differential utility for an infinite horizon when \(\theta <0\)

  • 期刊论文

  • 发表日期：

    2026年5月27日

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  • Yuki Shigeta

摘要

Recent studies highlight the difficulties in infinite-horizon Epstein–Zin stochastic differential utility from economic and mathematical perspectives when the coefficient of relative risk aversion and the elasticity of intertemporal substitution both exceed 1 (i.e., θ < 0 $\theta < 0$ ). We demonstrate that the economically problematic behaviour identified in recent studies disappears by an order-equivalent transformation of the utility index. Furthermore, we introduce an admissible set of consumptions to tackle mathematical issues for the existence of the utility. For applications, we examine a Merton problem and demonstrate that an optimal control derived from the Hamilton–Jacobi–Bellman equation is admissible and optimal under mild conditions.