Jensen Inequality of Bivariate Function in the G-Expectation Framework

Liyang Feng

doi:10.4236/jmf.2020.101004

Abstract

In the G-expectation framework, Wang [1] first obtained the Jensen inequality of one-dimensional function. In this paper, under some stronger conditions, we obtain the Jensen inequality of bivariate function based on Wang’s proof method. And we give some examples to illustrate the application of Jensen inequality of bivariate function.

Highlights

Open AccessAs we know, expected utility theory has been widely used in the field of mathematical finance, especially in measuring the preference and aversion of risk
We give some examples to illustrate the application of Jensen inequality of bivariate function
We can find that the Jensen inequality of bivariate function can be used to proof the inequality or estimate the G-expectation

Summary

Introduction

As we know, expected utility theory has been widely used in the field of mathematical finance, especially in measuring the preference and aversion of risk. Economists hope to find a tool that can have certain properties of the classical expectation and accurately measure risk aversion Driven by this problem, Peng [2] introduced a nonlinear expectation—g-expectation by backward stochastic differential equation in 1997 and in 2006, Peng [3] [4] [5] introduced a new nonlinear expectation—G-expectation through the nonlinear heat equation and established a systematical theoretical framework. In the G-expectation framework, Wang [1] studied the Jensen inequality of one-dimensional function under some sufficient and necessary conditions and illustrated the significant application of Jensen inequality in the. In this paper, based on Wang’s proof method, under some reasonable conditions, we obtain the Jensen inequality of bivariate function in the G-expectation framework. G-Jensen inequality of bivariate function under the stronger conditions and give some examples of Jensen inequality of binary function

Preliminaries and Notation

Bivariate Convex Function

Demonstrations

Conclusion