文章考虑有限期限上的最优投资消费问题. 风险资产服从几何布朗运动, 利率服从一个遍历的\ Markov 过程. 目标是累积消费和终值财富贴现的幂效用期望最大化. 利用动态规划原理推导出值函数所满足的\ HJB 方程, 并利用上下解方法证明了对应非线性抛物型偏微分方程终值问题解的存在唯一性, 最后证明了验证性定理.

In this paper, we consider an optimal investment and consumption problem on a finite time horizon. The price of risky asset obeys a geometric Brownian motion, and interest rate varies according to an ergodic Markov process. The goal is to choose optimal investment and consumption policies to maximize the expected discounted power utilities of the accumulative consumption and the terminal wealth. The {\rm HJB} equation is derived using dynamic programming principle, and the existence and uniqueness of solution of the terminal value problem for the corresponding nonlinear parabolic partial differential equation are obtained using the sub-supersolution method, finally, the verification theorem is obtained.