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

We study finite horizon optimal stopping problems for continuous time Feller-Markov processes. The functional depends on time, state and external parameters, and may exhibit discontinuities with respect to the time-variable. Both left and right-hand discontinuities are considered. We investigate the dependence of the value function on the parameters, initial state of the process and on the stopping horizon. We construct ε-optimal stopping times and provide conditions under which an optimal stopping time exists. We demonstrate how to approximate this 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.