Optimal consumption and portfolio selection with Epstein-Zin utility under general constraints

TL;DR

This paper characterizes optimal consumption and investment strategies for an Epstein-Zin utility investor in incomplete markets with general constraints, using quadratic BSDEs and advanced mathematical techniques.

Contribution

It introduces a novel approach to solving the utility maximization problem under complex constraints via quadratic BSDEs and Lyapunov functions, extending existing methods.

Findings

Explicit characterization of optimal strategies.

Numerical simulations illustrating strategy behavior.

Handling unbounded solutions in stochastic environments.

Abstract

The paper investigates the consumption-investment problem for an investor with Epstein-Zin utility in an incomplete market. Closed, not necessarily convex, constraints are imposed on strategies. The optimal consumption and investment strategies are characterized via a quadratic backward stochastic differential equation (BSDE). Due to the stochastic market environment, the solution to this BSDE is unbounded and thereby the BMO argument breaks down. After establishing the martingale optimality criterion, by delicately selecting Lyapunov functions, the verification theorem is ultimately obtained. Besides, several examples and numerical simulations for the optimal strategies are provided and illustrated.