G-expectation, G-Brownian motion and related Stochastic calculus of ito type

We introduce a notion of nonlinear expectation - G-expectation generated by a nonlinear heat equation with a given infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of lto's type with respect to our G-Brownian motion and derive the related Ito's formula. We have also given the existence and uniqueness of stochastic differential equation under our G-expectation. As compared with our previous framework of g-expectations, the theory of G-expectation is intrinsic in the sense that it is Dot based on a given (linear) probability space.