Weak dynamic programming principle for viscosity solutions

Abstract. We prove a weak version of the dynamic programming principle for standard stochas-tic control problems and mixed control-stopping problems, which avoids the technical difficulties related to the measurable selection argument. In the Markov case, our result is tailor-made for the derivation of the dynamic programming equation in the sense of viscosity solutions. Key words. Optimal control, Dynamic programming, discontinuous viscosity solutions. AMS subject classifications. Primary 49L25, 60J60; secondary 49L20, 35K55. 1. Introduction. Conside