On Quadratic g-Evaluations/Expectations and Related Analysis

Abstract

In this article we extend the notion of g-evaluation, in particular g-expectation, of Peng [8, 9] to the case where the generator g is allowed to have a quadratic growth (in the variable “z”). 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 upcrossing inequality. The results of this article are important in the further development of the general quadratic nonlinear expectations (cf. [5]).

Keywords:

• BMO
• Doob–Meyer decomposition
• Jensen's inequality
• Optional sampling
• Quadratic g-evaluations
• Quadratic g-expectations
• Reverse comparison theorem
• Upcrossing inequality

Mathematics Subject Classification:

• Primary 60G48
• Secondary 60H10, 91B30

We would like to express our sincere gratitude to the anonymous referee for the careful reading of our original manuscript and many valuable suggestions that helped us improve the quality of the paper significantly.

J.M. is supported in part by NSF grant #0505427.