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

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.