Consumption-portfolio optimization with recursive utility in incomplete markets

In an incomplete market, we study the optimal consumption-portfolio decision of an investor with recursive preferences of Epstein-Zin type. Applying a classical dynamic programming approach, we formulate the associated Hamilton-Jacobi-Bellman equation and provide a suitable verification theorem. The proof of this verification theorem is complicated by the fact that the Epstein-Zin aggregator is non-Lipschitz, so standard verification results (e.g. in Duffie and Epstein, Econometrica 60, 393-394, 1992) are not applicable. We provide new explicit solutions to the Bellman equation with Epstein-Zin preferences in an incomplete market for non-unit elasticity of intertemporal substitution (EIS) and apply our verification result to prove that they solve the consumption-investment problem. We also compare our exact solutions to the Campbell-Shiller approximation and assess its accuracy.