Dynamic Programming Principle for Classical and Singular Stochastic Control with Discretionary Stopping

We prove the dynamic programming principle (DPP) in a class of problems where an agent controls a d -dimensional diffusive dynamics via both classical and singular controls and, moreover, is able to terminate the optimisation at a time of her choosing, prior to a given maturity. The time-horizon of the problem is random and it is the smallest between a fixed terminal time and the first exit time of the state dynamics from a Borel set. We...