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

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].