Quantifying non-monotonicity of functions and the lack of positivity in signed measures

Abstract

In various research areas related to decision making, problems and their solutions frequently rely on certain functions being monotonic. In the case of non-monotonic functions, one would then wish to quantify their lack of monotonicity. In this paper we develop a method designed specifically for this task, including quantification of the lack of positivity, negativity, or sign-constancy in signed measures. We note relevant applications in Insurance, Finance, and Economics, and discuss some of them in detail.

Keywords

• Non-monotonic functions
• signed measures
• Hahn and Jordan decompositions
• weighted premium
• risk measure
• gain–loss ratio