Existence conditions of the optimal stopping time: the cases of geometric Brownian motion and arithmetic Brownian motion

A type of optimal investment problem can be regarded as an optimal stopping problem in the field of applied stochastic analysis. This study derives the existence conditions of the optimal stopping time when the stochastic process is a geometric Brownian motion or an arithmetic Brownian motion. The conditions concern the intrinsic value function and are natural extensions of the certainty case. Additionally, they are essential for a well-known result in recent investment theory. They are also applied to an optimal land development problem. The analyses give existing studies rigorous foundations and generalize them.