Abstract

Nonlinear expectation theory has significantly enlarged the foundation of probability theory established by A. N. Kolmogorov in 1933 through the nonlinearity of the notion of expectation. In this lecture we explain in details the state of arts of this new theory, the theoretical and statistical meaning as well as the mechanism hidden behind of nonlinear law of large numbers, the nonlinear central limit theorem as well as the nonlinear version of stochastic calculus, including backward stochastic differential equations and forward stochastic differential equations, their applications and some open problems.