On Quadratic g-Evaluations/Expectations and Related Analysis

In this paper we extend the notion of g-evaluation, in particular g-expectation, to the case where the generator g is allowed to have a quadratic growth. We show that some important properties of the g-expectations, including a representation theorem between the generator and the corresponding g-expectation, and consequently the reverse comparison theorem of quadratic BSDEs as well as the Jensen inequality, remain true in the quadratic case. Our main results also include a Doob-Meyer type decomposition, the optional sampling theorem, and the up-crossing inequality. The results of this paper are important in the further development of the general quadratic nonlinear expectations.