Time-Inconsistent Mean-Utility Portfolio Selection with Moving Target

In this paper, we solve the time inconsistent portfolio selection problem by using different utility functions with a moving target as our constraint. We solve this problem by finding an equilibrium control under the given definition as our optimal control. We firstly derive a sufficient equilibrium condition for second-order continuously differentiable utility funtions. Then we use power functions of order two, three and four in our problem and find the respective condtions for obtaining an equilibrium for our different problems. In the last part of the paper, we consider using another definition of equilibrium to solve our problem when the utility function that we use in our problem is the negative part of x and also find the condtions for obtaining an equilibrium.