Set-valued Star-Shaped Risk Measures

Paper: arXiv 2402.18014

Authors: Bingchu Nie, Dejian Tian, Long Jiang

Abstract

In this paper, we introduce a new class of set-valued risk measures, named set-valued star-shaped risk measures. Motivated by the results of scalar monetary and star-shaped risk measures, this paper investigates the representation theorems in the set-valued framework. It is demonstrated that set-valued risk measures can be represented as the union of a family of set-valued convex risk measures, and set-valued normalized star-shaped risk measures can be represented as the union of a family of set-valued normalized convex risk measures. The link between set-valued risk measures and set-valued star-shaped risk measures is also established.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly abstract mathematical framework extending star-shaped risk measures to set-valued spaces, featuring rigorous representation theorems. However, it is entirely theoretical with no empirical application, backtesting, or real-world financial datasets provided.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop Set-Valued Star-Shaped Risk Measures] --> B[Inputs: Scalar Monetary & Star-Shaped Risk Measure Theories]
    B --> C[Methodology: Extend Framework to Set-Valued Axioms]
    C --> D[Computational Process: Formulate Union of Set-Valued Convex Risk Families]
    D --> E[Outcome 1: Representation Theorem for General Set-Valued Risk Measures]
    D --> F[Outcome 2: Representation Theorem for Normalized Set-Valued Risk Measures]
    E & F --> G[Final Finding: Link Established Between Set-Valued Star-Shaped & Convex Risk Measures]