Consumption and portfolio optimization solvable problems with recursive preferences

Abstract

This paper considers the consumption and portfolio optimization problems with recursive preferences in both infinite and finite time regions, in which the financial market consists of a risk-free asset and a risky asset following a general stochastic volatility process. By using Bellman’s dynamic programming principle, the Hamilton–Jacobi–Bellman (HJB) equation is derived for characterizing the optimal consumption–investment strategy and the corresponding value function. Based on the conjecture of the exponential-polynomial form of the value function under mild conditions, we prove that, when the order of the polynomial $n\le 2$, the HJB equation has an analytical solution if the investor with unit elasticity of intertemporal substitution and an approximate solution by the log-linear approximation method otherwise. We also prove that the HJB equation has no solutions under the conjecture of the exponential-polynomial form of the value function when the order of the polynomial $n>2$. Finally, the optimal consumption–portfolio strategies to Heston’s model are provided and some numerical experiments are given to illustrate the behavior of the optimal consumption–portfolio strategies.