  Under the framework of G-expectation and G-Brownian motion, we introduce
It\^o's integral for stochastic processes without assuming quasi-continuity.
Then we can obtain It\^o's integral on stopping time interval. This new
formulation permits us to obtain It\^o's formula for a general
C^{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].
