In this paper, we investigate an interesting and important stopping problem mixed with stochastic controls and a textit{nonsmooth} utility over a finite time horizon. The paper aims to develop new methodologies, which are significantly different from
those of mixed dynamic optimal control and stopping problems in the existing literature, to figure out a managers decision. We formulate our model to a free boundary problem of a fully textit{nonlinear} equation. By means of a dual transformation, however, we can convert the above problem to a new free boundary problem of a textit{linear} equation. Finally, using the corresponding inverse dual transformation, we apply the theoretical results established for the new free boundary problem to obtain the properties of the optimal strategy and the optimal stopping time to achieve a certain level for the original problem over a finite time investment horizon.
In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic op
timal control and stopping problems in the existing literature, to study a managers decision. We formulate our model to a free boundary problem of a fully nonlinear equation. Furthermore, by means of a dual transformation for the above problem, we convert the above problem to a new free boundary problem of a linear equation. Finally, we apply the theoretical results to challenging, yet practically relevant and important, risk-sensitive problems in wealth management to obtain the properties of the optimal strategy and the right time to achieve a certain level over a finite time investment horizon.
