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We aim to give an overview on how to derive the dynamic programming principlefor a general stochastic control/stopping problem, using measurable selectiontechniques. By considering their martingale problem formulation, we show how tocheck the required measurability conditions for different versions ofcontrol/stopping problem, including in particular the controlled/stoppeddiffusion processes problem. Further, we also study the stability of thecontrol problem, i.e. the approximation of the control process by piecewiseconstant control problems. It induces in particular an equivalence result fordifferent versions of the controlled/stopped diffusion processes problem.

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