  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 optimal control and stopping problems in the existing
literature, to study a manager's 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.
