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We prove the dynamic programming principe for uniformly nondegeneratestochastic differential games in the framework of time-homogeneous diffusionprocesses considered up to the first exit time from a domain. In contrast withprevious results established for constant stopping times we allow arbitrarystopping times and randomized ones as well. There is no assumption aboutsolvability of the the Isaacs equation in any sense (classical or viscosity).The zeroth-order "coefficient" and the "free" term are only assumed to bemeasurable in the space variable. We also prove that value functions areuniquely determined by the functions defining the corresponding Isaacsequations and thus stochastic games with the same Isaacs equation have the samevalue functions.

On the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the Isaacs equations