Kraft, Holger2; Seifried, Frank Thomas3; Steffensen, Mogens4
1 Department of Mathematical Sciences, Faculty of Science, Københavns Universitet2 Department of Finance, Goethe University, Frankfurt am Main, Germany3 Department of Mathematics, University of Kaiserslautern, Germany4 Department of Mathematical Sciences, Faculty of Science, Københavns Universitet
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.