This paper revisits the classical Merton problem on the finite horizon with the constant absolute risk aversion utility function. We apply two different methods to derive the closed-form solution of the corresponding Hamilton–Jacobi–Bellman (HJB) equation. An approximating method consists of two steps: solve the HJB equation with the hyperbolic absolute risk aversion utility function first and then take the limits of the risk aversion parameter to negative infinite. A direct method is also provided to derive another closed-form solution. Finally, we prove that the solutions obtained from different methods are equivalent. In addition, a sufficient condition is proposed to guarantee the optimal consumption is nonnegative and such a condition also leads to the verification theorem. A great advantage of our derived solution is that optimal policies can now be quantitatively scrutinized and discussed from both mathematical and economic viewpoints.

本文用恒定的绝对风险规避效用函数在有限范围内回顾了经典的默顿问题。我们应用两种不同的方法来推导相应的汉密尔顿-雅各比-贝尔曼（HJB）方程的闭式解。一种近似方法包括两个步骤：首先用双曲绝对风险规避效用函数求解HJB方程，然后将风险规避参数的极限取为负无穷大。还提供了一种直接方法来导出另一个封闭形式的解决方案。最后，我们证明了从不同方法获得的解是等效的。另外，提出了充分的条件以保证最优消耗是非负的，并且这种条件也导致验证定理。