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

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