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. The article considers the optimal investment and consumption problem over a finite period.The risky asset obeys the geometric Brownian motion and the interest rate obeys a traversal Markov process.The goal is to maximize the exponential utility of the cumulative consumption and the final wealth discounting.Using the principle of dynamic programming, Function is satisfied, and the existence and uniqueness of the solution to the final value problem of the corresponding nonlinear parabolic partial differential equation are proved by using the upper and lower solutions. Finally, the confirmatory theorem is proved.