Jensen's inequality for monetary utility functions

Jing Liu,Long Jiang

doi:10.1186/1029-242x-2012-128

Jing Liu, Long Jiang

Open Access

https://doi.org/10.1186/1029-242x-2012-128

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Abstract

In this paper, we prove that Jensen's inequality holds true for all monetary utility functions with respect to certain convex or concave functions by studying the properties of monetary utility functions, convex functions and concave functions.

Highlights

Introduction and preliminaries1.1 Introduction Monetary utility functions have recently attracted much attention in the mathematical finance community, see e.g. [1,2,3,4]
According to [2], a monetary utility function U can be identified with a convex risk measure r by the formula U(ξ) = - r(ξ); convex risk measure, introduced in [5,6], is a popular notion in particular since the Basel II accord
With respect to some convex or concave functions, it does not hold true for all monetary utility functions, as stated in our Example 2.1

Summary

Introduction

Introduction and preliminaries1.1 Introduction Monetary utility functions have recently attracted much attention in the mathematical finance community, see e.g. [1,2,3,4]. 1.1 Introduction Monetary utility functions have recently attracted much attention in the mathematical finance community, see e.g. It is well-known that Jensen’s inequality holds true for classical expectation, which, in terms of operator, can be seen as a particular type of monetary utility functions. With respect to some convex or concave functions, it does not hold true for all monetary utility functions, as stated in our Example 2.1.

Results

Conclusion