投资和消费,是人们日常生活中再平凡不过的行为。对于连续时间的模型,通常用几何布朗运动来描述风险资产价格的变动规律,投资者需要对投资和消费进行控制,来最大化累积消费效用的期望值。本文利用伊藤公式,得到一个关于目标函数的完全非线性的常微分方程,然后通过对偶变换的方法,将方程转化为欧拉方程,从而给出解的表达式。 Investment and consumption are commonplace behaviors in people’s daily life. For continuous-time models, the geometric Brownian motion is usually used to describe the variation of the price of risky assets. Investors need to control their investment and consumption to maximize the expected value of cumulative consumption utility. In this paper, we use Ito’s formula to obtain a completely nonlinear ordinary differential equation about the objective function, and then transform the equation into the Euler’s equation through the method of dual transformation, so as to give the expression of the solution.