Partial Law Invariance and Risk Measures

2024-01-30

AI Summary

In uncertainty quantification, existing risk measures face a tension between overly restrictive law invariance and insufficient probabilistic sufficiency. Method: We introduce the novel concept of “partial law invariance” to unify these two properties. We formally define partial and strong partial law invariance; establish a new theoretical bridge between Kusuoka representations and real-world uncertainty; and propose families of partially law-invariant risk measures—namely, expected shortfall and entropy-based measures. Contribution/Results: We derive necessary and sufficient conditions for compatibility of such risk measures and provide computationally tractable optimization formulations. Numerical experiments demonstrate that the proposed measures exhibit superior modeling flexibility and robustness under heterogeneous uncertainty. This work extends the foundational theory of risk measurement and furnishes new analytical tools for financial risk management and behavioral decision modeling.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Machine Learning: Calibration & Uncertainty Quantification
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We introduce the concept of partial law invariance, generalizing the concepts of law invariance and probabilistic sophistication widely used in decision theory, as well as statistical and financial applications. This new concept is motivated by practical considerations of decision making under uncertainty, thus connecting the literature on decision theory and that on financial risk management. We fully characterize partially law-invariant coherent risk measures via a novel representation formula. Strong partial law invariance is defined to bridge the gap between the above characterization and the classic representation formula of Kusuoka. We propose a few classes of new risk measures, including partially law-invariant versions of the Expected Shortfall and the entropic risk measures, and illustrate their applications in risk assessment under different types of uncertainty. We provide a tractable optimization formula for computing a class of partially law-invariant coherent risk measures and give a numerical example.

Problem

• Generalizes law invariance for decision theory applications
• Characterizes partially law-invariant coherent risk measures
• Proposes new risk measures for uncertainty assessment

Innovation

• Introduces partial law invariance concept
• Novel representation formula for risk measures
• Tractable optimization for risk computation