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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. The zeroth-order"coefficient" and the "free" term are only assumed to be measurable. Incontrast with previous results established for constant stopping times we allowarbitrary stopping times and randomized ones as well. The main assumption,which will be removed in a subsequent article, is that there exists asufficiently regular solution of the Isaacs equation.

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