This thesis focuses on theoretically investigating the free boundary problems which arise from two classical problems in mathematical finance: portfolio selection and optimal contract of principal-agent theory. In these two problems, the mathematical structure is not represented as a solely optimal stopping time problem but is formulated in a mixture form in which optimal stopping time is combined with stochastic control or singular control. In this structure, the characterization of the free boundary for optimal stopping time is subtle and the verification theorem of the optimality for the candidate control and stopping time is difficult to prove. To overcome these difficulties, we utilize the dual/martingale method, and then we analyze the variational inequality arising from the dual problem and formally provide the derivation of the solution to the variational inequality. In the thesis, we make a technical contribution by fully characterizing the free boundary and providing the duality and verification theorems. Based on the analytic results, we seek to gain further insight into two classical problems under several realistic model setups.