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Under the framework of G-expectation and G-Brownian motion, we introduceIt\^o's integral for stochastic processes without assuming quasi-continuity.Then we can obtain It\^o's integral on stopping time interval. This newformulation permits us to obtain It\^o's formula for a generalC^{1,2}-function, which essentially generalizes the previous results of Peng[20, 21, 22, 23, 24] as well as those of Gao [8] and Zhang et al. [26].

Stopping Times and Related It\^o's Calculus with G-Brownian Motion