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 Ito's type with respect to Our G-Brownian motion, and derive the related Ito's formula. We have also obtained the existence and uniqueness of stochastic differential equations under Our G-expectation. (C) 2008 Elsevier B.V. All rights reserved.