Paper: arXiv 2407.18687

Authors: Marcelo Righi, Eduardo Horta, Marlon Moresco

Abstract

We introduce the concept of set risk measures (SRMs), which are real-valued maps defined on the space of all non-empty, closed, and bounded sets of almost surely bounded random variables. Traditional risk measures typically operate on random variables, but SRMs extend this framework to sets of random variables. We establish an axiomatic scheme for SRMs, similar to classical risk measures but adapted for set operations. The main technical contribution is an axiomatic dual representation of convex SRMs by using regular, $τ$-additive (for nets) measures on the unit ball of the dual space of essentially bounded random variables. We explore worst-case SRMs, which evaluate risk as the supremum of individual risks within a set, and provide a collection of examples illustrating the applicability of our framework to systemic risk, optimization, and decision-making under uncertainty.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical foundation for set risk measures using advanced functional analysis, dual representations, and hypertopologies. However, it is purely theoretical with no empirical backtesting or real-world financial data validation included.

Research Flowchart

  flowchart TD
    A["Research Goal: Extend risk measures to sets of random variables (SRMs)"] --> B["Input: Non-empty, closed, bounded sets in L^∞"]
    B --> C["Axiomatic Framework: Adapt classical risk axioms to set operations"]
    C --> D["Dual Representation: Apply regular τ-additive measures on dual unit ball"]
    C --> E["Worst-Case Analysis: Evaluate risk via supremum of set elements"]
    D --> F["Outcomes: Duality theory & applications to systemic risk & decision-making"]
    E --> F

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1 of 53 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.