Capacities, Measurable Selection and Dynamic Programming Part II: Application in Stochastic Control Problems

Abstract

We provide an overview on how to use the measurable selection techniques to derive the dynamic programming principle for a general stochastic optimal control/stopping problem. By considering its martingale problem formulation on the canonical space of paths, one can check the required measurability conditions. This covers in particular the most classical controlled/stopped diffusion processes problems. Further, we study the approximation property of the optimal control problems by piecewise constant control problems. As a byproduct, we obtain an equivalence result of the strong, weak and relaxed formulations of the controlled/stopped diffusion processes problem.