Explicit solutions to some optimal variance stopping problems

The stopping problem with variance as the optimality criterion is introduced. Due to the variance criterion, smooth fit cannot be applied directly. The problem is solved by embedding it into tractable auxiliary optimal stopping problems, where smooth fit is used to obtain explicit, optimal solutions. Optimal strategies are presented in closed form for several examples. A characteristic feature is that the optimal stopping boundaries depend on the initial value of the gain process, i.e. the state space of the gain process does not split into one continuation set and one stopping set.