We will briefly introduce various types of stochastic control
 problems, their connections with probability and PDE, and their
 applications in mathematical finance. As a specific example,
 we consider a stochastic control problem with stopping time
 in a finite time horizon. In general framework, the value function,
 required to be continuous, is characterized as the
 unique viscosity solution. However, the necessary and sufficient
 conditions for continuity remain unclear on a bounded domain.
 We will discuss the possibility to generalize the existing result on
 sufficient conditions for the continuity by observing a
 sample path local behavior of the underlying random processes.