G-Expectation, G-Brownian Motion And Related Stochastic Calculus Of It(?) Type

In this paper, we introduce a notion of nonlinear expectation—G-expextation—generated by a nonlinear hear equation. Different from any other "expectation", the G-expectation is not based on a given (linear) probability space. We first discuss the notion of G-normal distribution. With this nonlinear distribution we introduce G-expectation under which the canonical process is G-brownian motion. We then intro-duce the related stochastic integrals of Ito's type with respect to G-Brownian Motion. At last, we briefly introduce the law of large numbers and central limit theory under nonlinear expectations.