On $g$-expectations and filtration-consistent nonlinear expectations

In this paper, we obtain a comparison theorem and a invariant representation theorem for backward stochastic differential equations (BSDEs) without any assumption on the second variable $z$. Using the two results, we further develop the theory of $g$-expectations. Filtration-consistent nonlinear expectation ($\mathcal{F}$-expectation) provides an ideal characterization for the dynamical risk measures, asset pricing and utilities. We propose two new conditions: an absolutely continuous condition and a (locally Lipschitz) domination condition. Under the two conditions respectively, we prove that any $\mathcal{F}$-expectation can be represented as a $g$-expectation. Our results contain a representation theorem for $n$-dimensional $\mathcal{F}$-expectations in the Lipschitz case, and two representation theorems for $1$-dimensional $\mathcal{F}$-expectations in the locally Lipschitz case, which contain quadratic $\mathcal{F}$-expectations.