Stopping of functionals with discontinuity at the boundary of an open set

We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set $\mathcal{O}$. The stopping horizon is either random, equal to the first exit from the set $\mathcal{O}$, or fixed: finite or infinite. The payoff function is continuous with a possible jump at the boundary of $\mathcal{O}$. Using a generalization of the penalty method we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or $ϵ$-optimal stopping times.