Jensen's Inequality for g-Convex Function under g-Expectation

TL;DR

This paper characterizes g-convex functions under g-expectation by providing necessary and sufficient conditions for C^2 functions to satisfy a generalized Jensen's inequality, extending the understanding of g-convexity, g-concavity, and g-affinity.

Contribution

It offers a complete characterization of g-convex functions under g-expectation, including necessary and sufficient conditions for C^2 functions, and explores g-concave and g-affine functions.

Findings

Established necessary and sufficient conditions for g-convexity of C^2 functions.

Extended the analysis to g-concave and g-affine functions.

Provided a framework for understanding generalized Jensen's inequalities under g-expectation.

Abstract

A real valued function defined on}$\mathbb{R}$ {\small is called}$g${\small --convex if it satisfies the following \textquotedblleft generalized Jensen's inequality\textquotedblright under a given}$g${\small -expectation, i.e., }$h(\mathbb{E}^{g}[X])\leq \mathbb{E}% ^{g}[h(X)]${\small, for all random variables}$X$ {\small such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient conditions for a }$C^{2}${\small -function being}$% g ${\small -convex. We also studied some more general situations. We also studied}$g${\small -concave and}$g${\small -affine functions.