Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

We develop a notion of nonlinear expectation– G -expectation–generated by a nonlinear heat equation with infinitesimal generator G . We first study multi-dimensional G -normal distributions. With this nonlinear distribution we can introduce our G -expectation under which the canonical process is a multi-dimensional 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 obtained the existence and uniqueness of stochastic differential equations under our G -expectation.