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

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

2008-02-04

arxiv.org/abs/0802.0373v1

Mathematics

Probability

21 pages

Scientific paper

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.

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