Abstracts

In this talk we present a general solution of the optimal consumption and portfolio selection problem for an investor with recursive preferences of Epstein-Zin type in an incomplete market. Analytic solutions for special parameterizations have previously been obtained by Chacko and Viceira (2005) and Kraft, Seifried and Steffensen (2013). We approach the optimization problem via the associated Hamilton-Jacobi-Bellman (HJB) partial differential equation. First it is shown that solutions of the HJB equation that satisfy a boundedness condition provide the solution to the corresponding consumption-portfolio optimization problem. For this novel verification theorem for SDU in incomplete markets, utility gradient inequalities similar to those of Schroder and Skiadas (1999) are used in combination with HJB methods. Finally we employ a fixed point argument to construct a classical solution of the HJB meeting the required boundedness conditions. More precisely, generalizing the approach of Berdjane and Pergamenshchikov (2013), we study the Feynman-Kac representation mapping $\Phi$ that is associated to a power transform of the HJB equation. A fixed point argument yields a fixed point of $\Phi$ in the space of continuous functions as a limit of iterations of $\Phi$. Using the probabilistic representation of this solution we are able to deduce convergence in $C^{0,1}$. This not only yields a theoretical optimality result, but also leads directly to an efficient method for the numerical computation of optimal strategies by iteratively solving linear parabolic PDEs. Our proposed method exhibits superlinear convergence. We illustrate our results for various popular models, including the Heston stochastic volatility model.