Infinite Horizon Optimal Consumption: Intertemporal Hedging under Epstein-Zin Preferences

2026-06-01

AI Summary

This study addresses the optimal consumption and investment problem for an investor with Epstein–Zin stochastic differential utility in incomplete markets over an infinite horizon, allowing for stochastic investment opportunities and disentangling risk aversion from the elasticity of intertemporal substitution. Employing a variational approach, the paper establishes—for the first time—the variational representation of Epstein–Zin utility for any nonnegative adapted consumption stream when the preference parameter θ lies in (0,1), thereby overcoming conventional convexity restrictions. The value function is characterized as the unique minimizer of a certain functional, and a verification theorem is developed through measure changes, uniqueness results for backward stochastic differential equations, and perturbation arguments. The resulting minimizer is shown to be strictly positive and classical, yielding a feedback-form optimal policy, whose ability to capture intertemporal hedging behavior is demonstrated in models featuring stochastic volatility and heavy-tailed returns.

Abstract

We study an infinite-horizon optimal consumption-investment problem for an investor with Epstein-Zin stochastic differential utility with stochastic investment opportunities in an incomplete market. Risk aversion and intertemporal substitution are separated, and we work in the regime $θ\in(0,1)$, where there exists a unique generalised utility process for arbitrary non-negative progressively measurable consumption streams. Our main contribution is a variational characterisation of the value function. We show that the value function is the unique minimiser of a functional whose Euler-Lagrange equation coincides with the Hamilton-Jacobi-Bellman equation. Although the functional may be non-convex, the direct method yields existence, and we prove every minimiser is strictly positive, bounded, and classical. A verification theorem identifies any minimiser with the value function and gives feedback representations for optimal consumption and investment policies. The proof combines a change of measure to the myopic probability with uniqueness results for Epstein-Zin BSDEs and a perturbation argument for optimality. Examples with stochastic volatility, Gaussian excess returns, and fat-tailed excess returns illustrate the scope of the framework and its implications for intertemporal hedging.

Problem

• Optimal consumption
• Epstein-Zin preferences
• Incomplete markets
• Stochastic investment opportunities
• Intertemporal hedging

Innovation

• Epstein-Zin preferences
• variational characterization
• Hamilton-Jacobi-Bellman equation
• stochastic differential utility
• intertemporal hedging