Representation Theorems for Quadratic F-Consistent Nonlinear Expectations

In this paper we extend the notion of “filtration-consistent nonlinear expectation” (or “F-consistent nonlinear expectation”) to the case when it is allowed to be dominated by a g-expectation 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 [J. Ma, S. Yao, Quadratic g-evaluations and g-martingales, 2007, preprint]. The main contribution of this paper is the finding of a domination condition to replace the one used in all the previous works (e.g., [F. Coquet, Y. Hu, J. M´emin, S. Peng, Filtration-consistent nonlinear expectations and related g-expectations, Probab. Theory Related Fields 123 (1) (2002) 1–27; S. Peng, Nonlinear expectations, nonlinear evaluations and risk measures, in:Stochastic Methods in Finance, in: Lecture Notes in Math., vol. 1856, Springer, Berlin, 2004, pp. 165–253]), which is no longer valid in the quadratic case.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