Consumption and portfolio optimization solvable problems with recursive preferences

This paper considers consumption and portfolio optimization problems with recursive preferences in both infinite and finite time regions. Specially, the financial market consists of a risk-free asset and a risky asset that follows 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, we prove that, when the order of the polynomial $n\leq2$, the HJB equation has an analytical solution if the investor with unit elasticity of intertemporal substitution (EIS) and an approximate solution otherwise.