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

We develop a notion of nonlinear expectation–GG-expectation–generated by a nonlinear heat equation with infinitesimal generator GG. We first study multi-dimensional GG-normal distributions. With this nonlinear distribution we can introduce our GG-expectation under which the canonical process is a multi-dimensional GG-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our GG-Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our GG-expectation.