Optimal stopping problems with restricted stopping times

This paper provides a general ground for the problems of optimal stopping times over the families of partially available (or restricted) stopping times. It subsumes the classical framework in continuous-time, discrete-time, as well as semi-Markov settings as special cases. We model the problem by a restricted pool of stopping times meeting certain natural conditions and present its solution by means of Snell's envelope technique that extends the classical results. We further extend this type of problems to the stochastic processes indexed by partially ordered set.