2 00 7 Representation Theorems for Quadratic F - Consistent Nonlinear Expectations ∗

In this paper we extend the notion of “filtration-consistent nonlinear expectation” (or “Fconsistent nonlinear expectation”) to the case when it is allowed to be dominated by a gexpectation that may have a quadratic growth. We show that for such a nonlinear expectation many fundamental properties of a martingale can still make sense, including the Doob-Meyer type decomposition theorem and the optional sampling theorem. More importantly, we show that any quadratic F-consistent nonlinear expectation with a certain domination property must be a quadratic g-expectation as was studied in Ma-Yao [11]. The main contribution of this paper is the finding of the domination condition to replace the one used in all the previous works (e.g., [6] and [14]), which is no longer valid in the quadratic case. We also show that the representation generator must be deterministic, continuous, and actually must be of the simple form g(z) = μ(1 + |z|)|z|, for some constant μ > 0.