G-Martingales and Jensen’s Inequality

Abstract

In this chapter, we introduce the notion of G-martingales and the related Jensen’s inequality for a new type of G-convex functions. One essential difference from the classical situation is that here “M is a G-martingale” does not imply that “$-M$ is a G-martingale”.

Notes and Comments

The material in this chapter is mainly from Peng [140].

Peng [130] introduced a filtration consistent (or time consistent, or dynamic) nonlinear expectation, called g-expectation, via BSDE, developed further in (1999) [132] for some basic properties of the g-martingale such as nonlinear Doob-Meyer decomposition theorem. See also Briand et al. [20]  , Chen et al.   [29] , Chen and Peng [30, 31], Coquet, Hu, Mémin and Peng [35, 36], Peng [132, 135], Peng and Xu [148], Rosazza   [152]. These works lead to a conjecture that all properties obtained for g-martingales must have their counterparts for G-martingale. However this conjecture is still far from being complete.

The problem of G-martingale representation has been proposed by Peng [140]. In Sect. 4.2, we only state a result with very regular random variables. Some very interesting developments to this important problem will be provided in Chap. 7.

Under the framework of g-expectation, Chen, Kulperger and Jiang [29], Hu [86], Jiang and Chen [97] investigate the Jensen’s inequality for g-expectation. Jia and Peng [95] introduced the notion of g-convex function and obtained many interesting properties. Certainly, a G-convex function concerns fully nonlinear situations.