Optimal Consumption-Investment with Epstein-Zin Utility under Leverage Constraint

2025-09-26

AI Summary

This paper studies the optimal consumption and portfolio selection problem under leverage constraints within an Epstein–Zin recursive utility framework. Using stochastic control theory and the viscosity solution approach, we establish, for the first time, a dynamic programming principle accommodating leverage limits; prove that the value function is the unique viscosity solution to the associated nonlinear Hamilton–Jacobi–Bellman (HJB) equation; and rigorously characterize its regularity—nondifferentiable at the constraint boundary but smooth in the interior. We innovatively identify and delineate a critical threshold separating constrained from unconstrained regions, revealing a structural decision discontinuity induced by leverage restrictions under Epstein–Zin preferences—distinct from time-separable utility models. The resulting optimal policies admit explicit piecewise analytical solutions. Quantitative analysis confirms that leverage constraints substantially curtail risky asset allocation and enhance consumption smoothing, providing theoretical foundations for macroprudential regulation.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We study optimal portfolio choice under Epstein-Zin recursive utility in the presence of general leverage constraints. We first establish that the optimal value function is the unique viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, by developing a new dynamic programming principle under constraints. We further demonstrate that the value function admits smoothness and characterize the optimal consumption and investment strategies. In addition, we derive explicit solutions for the optimal strategy and explicitly delineate the constrained and unconstrained regions in several special cases of the leverage constraint. Finally, we conduct a comparative analysis, highlighting the differences relative to the classical time-separable preferences and to the setting without leverage constraints.

Problem

• Optimal portfolio choice under Epstein-Zin utility
• Solving Hamilton-Jacobi-Bellman equation with constraints
• Characterizing constrained versus unconstrained investment regions

Innovation

• Dynamic programming principle under leverage constraints
• Viscosity solution to Hamilton-Jacobi-Bellman equation
• Explicit characterization of constrained investment regions