G–Expectation, G–Brownian Motion and Related Stochastic Calculus of Itô 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 Itô’s type with respect to our G–Brownian motion and derive the related Itô’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 not based on a given (linear) probability space.