The notion of g-expectation was first introduced in 1993 by S. Peng shortly after the debut of theory on nonlinear backward stochastic differential equations. It has then been developed quite extensively, mainly due to its connections to the theory of risk measures, especially the dynamic coherent risk measures. The most amazing result obtained so far is that any regular coherent risk measure satisfying a certain "domination condition" must be a g-expectation, with g being Lipschitz continuous. Unfortunately, to date the representation results of this kind exclude an importance class of convex risk measures, including the ubiquitous entropic risk measures.

In this talk we show that the representation theorem can still hold for a fairly large class of convex risk measures, as long as they have at most quadratic growth. We introduce the notion of quadratic g-expectation as well as quadratic nonlinear expectations, using the recent development of the quadratic BSDEs. However, the extension gets surprisingly subtle due to the complete failure of the original domination condition. We show how this difficulty can be overcome by using the BMO theory, especially a so-called "Reversed Holder Inequality".

This talk is based on joint works with Y. Hu, S. Peng, and S. Yao.