Dynamic star-shaped risk measures and $g$-expectations

Paper: arXiv 2305.02481

Authors: Dejian Tian, Xunlian Wang

Abstract

Motivated by the results of static monetary or star-shaped risk measures, the paper investigates the representation theorems in the dynamic framework. We show that dynamic monetary risk measures can be represented as the lower envelope of a family of dynamic convex risk measures, and normalized dynamic star-shaped risk measures can be represented as the lower envelope of a family of normalized dynamic convex risk measures. The link between dynamic monetary risk measures and dynamic star-shaped risk measures are established. Besides, the sensitivity and time consistency problems are also studied. A specific normalized time consistent dynamic star-shaped risk measures induced by $ g $-expectations are illustrated and discussed in detail.

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 rigorous theoretical exploration of dynamic star-shaped risk measures and g-expectations using advanced stochastic calculus and mathematical finance frameworks. However, it is purely theoretical and completely lacks empirical backtesting, real-world market data validation, or implementation code.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Static Monetary & Star-Shaped Risk Measures to Dynamic Framework] --> B[Input: Financial Risk Models & BSDEs / g-Expectations]
    B --> C[Methodology: Derive Lower Envelope Representations via Dynamic Convex Risk Measures]
    C --> D[Analyze Properties: Dynamic Sensitivity & Time Consistency]
    D --> E[Computational Process: Construct Star-Shaped Risk Measures via g-Expectations]
    E --> F[Outcome: Established Link Between Dynamic Monetary & Star-Shaped Risk Measures]