An optimal investment problem with nonsmooth and nonconcave utility over a finite time horizon

In this paper, we study a class of optimal investment problems with a nonsmooth and nonconcave utility function, where the value function is the expected utility determined by the state process and time. We adopt partial differential equation methods to prove that the value function belongs to C^2,1 under some proper conditions of the utility function. Moreover, we analyze the continuity of the optimal strategy and obtain some of its properties around the boundary and the terminal time. Also, an example sheds light on the theoretical results established.