  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.
