Optimal stopping problems with constraints for continuous-time general Markov processes

In this work, we consider an optimal stopping time problem with a restriction that requires the stopping times to occur only when a given signal is observed. The dynamic of the stopping problem evolves as a continuous-time Markov process whose state space is a subset of a Polish space, whereas the signal is modeled by a piecewise deterministic Markov process, both processes are assumed to be independent of each other. To solve the problem, we employ the well-known discrete-time optimal control theory. In particular, we characterize the value function as the solution of a certain dynamic programming equation related to the discrete-time model and demonstrate the existence of an optimal stopping time through this equation. Our technique extends previous works that use more restrictive conditions on the state space of the process. As a byproduct, we intend to solve optimal stopping problems for a wider family of Markov processes.