Jensen's inequality for g-convex function under g-expectation

Abstract

A real valued function h defined on ℝ is called g-convex if it satisfies the "generalized Jensen's inequality" for a given g-expectation, i.e., h(Eg[X]) ≤ h(Eg[h(X)]) holds for all random variables X such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient condition for a C2-function being g-convex, and study some more general situations. We also study g-concave and g-affine functions, and a relation between g-convexity and backward stochastic viability property. © Springer-Verlag 2009.