In this paper, we obtain a comparison theorem and a invariant representation
theorem for backward stochastic differential equations (BSDEs) without any
assumption on the variable z. Using the two results, we further develop the
theory of g-expectations. Filtration-consistent nonlinear expectation
(F-expectation) provides an ideal characterization for the dynamical
risk measures, asset pricing and utilities. Under an absolutely continuous
condition and a domination condition, respectively, we prove that any
F-expectation can be represented as a g-expectation. Our results
contain a representation theorem for n-dimensional F-expectations
in the Lipschitz case, and two representation theorems for 1-dimensional
F-expectations in the locally Lipschitz case, which contain quadratic
F-expectations.Comment: 30 pages. This version corrects an error in the definition of a
stopping time in the proof of Theorem 2.7. Comments are welcom