Stopping times and related Itô’s calculus with -Brownian motion

Under the framework of G -expectation and G -Brownian motion, we introduce Itô’s integral for stochastic processes without assuming quasi-continuity. Then we can obtain Itô’s integral on stopping time interval. This new formulation permits us to obtain Itô’s formula for a general C 1 , 2 -function, which essentially generalizes the previous results of Peng (2006, 2008, 2009, 2010, 2010) [21–25] as well as those of Gao (2009) [8] and Zhang et al. (2010) [27].