A stochastic control problem and related free boundaries in finance

In this paper, we investigate an optimal stopping problem (mixed with stochastic controls) for a manager whose utility is nonsmooth and nonconcave over a finite time horizon. The paper aims to develop a new methodology, which is significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, so as to figure out the manager’s best strategies. The problem is first reformulated into a free boundary problem with a fully nonlinear operator. Then, by means of a dual transformation, it is further converted into a free boundary problem with a linear operator, which can be consequently tackled by the classical method. Finally, using the inverse transformation, we obtain the properties of the optimal trading strategy and the optimal stopping time for the original problem.