Using the linear programming approach to stochastic control introduced by Buckdahn, Goreac, and Quincampoix, and by Goreac and Serea, we provide a semigroup property for some set of probability measures leading to dynamic programming principles for stochastic control problems. An abstract principle is provided for general bounded costs. Linearized versions are obtained under further (semi)continuity assumptions.
"A note on linearization methods and dynamic programming principles for stochastic discontinuous control problems." Electron. Commun. Probab. 17 1 - 12, 2012. https://doi.org/10.1214/ECP.v17-1844