Finite Horizon Optimal Stopping of Discontinuous Functionals with Applications to Impulse Control with Delay∗

We study finite horizon optimal stopping problems for continuous time Feller-Markov pro-cesses. The functional depends on time, state of the process and external parameters, and may exhibit discontinuities with respect to the time-variable. Both left and right-hand discontinu-ities are considered. We investigate the dependence of the value function on the parameters, initial state of the process and on the stopping horizon. We show that an optimal stopping time exists and can be explicitly constructed once the value function is computed. We demonstrate how to approximate the optimal stopping time by solutions to discrete-time problems. Our results are applied to the study of impulse control problems with finite time horizon, decision lag and execution delay