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 OO. The stopping horizon is either random, equal to the first exit from the set OO, or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of OO. 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 εε-optimal stopping times.

► Optimal stopping of Feller–Markov processes with discontinuous functionals.
► Properties of the value function and existence of optimal stopping times.
► A generalization of the penalty method.
► Numerical algorithm for approximation of the value function and optimal stopping times.