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

Abstract: 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 $\epsilon$-optimal stopping times.