Optimal starting-stopping problems for Markov-Feller processes

By means of nested inequalities in semigroup form we give a characterization of the value functions of the starting-stopping problem for general Markov-Feller processes. Next, we consider two versions of constrained problems on the final state or on the final time. The optimal stopping problems have been extensively studied for diffusion processes, or other Markov processes, or for more general stochastic processes.