Paper: arXiv 2505.19276

Authors: Vasily Melnikov

Abstract

We introduce a new paradigm for risk sharing that generalizes earlier models based on discrete agents and extends them to allow for sharing risk within a continuum of agents. Agents are represented by points of a measure space and have potentially heterogeneous risk preferences modeled by risk measures on a separable probability space. We derive the dual representation of the value function using a Strassen-type theorem for the weak-star topology and provide a characterization of the acceptance set using Aumann integration. These results are illustrated by explicit formulas when risk preferences are within the family of entropic and expected shortfall risk measures, and applications to Pareto efficiency in large markets.

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 provides a rigorous mathematical generalization of risk-sharing theory from finite agents to a continuum using Gelfand integration, functional analysis, and weak-star Strassen-type theorems. However, it is purely theoretical with no empirical backtesting, datasets, or numerical simulations included.

Research Flowchart

  flowchart TD
    A[Goal: Generalize Risk Sharing to a Continuum of Agents] --> B[Inputs: Measure Space of Agents & Heterogeneous Risk Measures]
    B --> C[Derive Dual Value Function via Strassen-type Theorem]
    B --> D[Characterize Acceptance Sets using Aumann Integration]
    C --> E[Compute Explicit Formulas: Entropic & Expected Shortfall]
    D --> E
    E --> F[Outcomes: Proof of Pareto Efficiency in Large Markets]

Cited References (37)

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  3. Scenario-Based Risk Evaluation (2021)
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  5. Comonotone Pareto Optimal Allocations for Law Invariant Robust Utilities on $ L^ 1 $ (2014)
  6. Infinite Dimensional Analysis (2006)
  7. Measurability in a Banach Space (1977)
  8. On Convex Risk Measures on $ L^ p $-Spaces (2009)
and 29 more
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  3. Values of Large Games, III : A Corporation With Two Large Stockholders (1961)
  4. Solvency II , or How To Sweep the Downside Risk Under the Carpet (2018)
  5. Risk Sharing for Capital Requirements With Multidimensional Security Markets (2019)
  6. Optimal Capital and Risk Transfers for Group Diversification (2007)
  7. Allocation of Risks and Equilibrium in Markets With Finitely Many Traders (2008)
  8. Regulatory Arbitrage of Risk Measures (2016)
  9. Pareto-Optimal Risk Exchanges and Related Decision Problems (1978)
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  15. Maximum Lebesgue Extension of Monotone Convex Functions (2014)
  16. Quantile-Based Risk Sharing (2018)
  17. On the Pettis Integral of Closed Valued Multifunctions (2000)
  18. On the Strassen Theorem (1974)
  19. Stochastic Finance (2002)
  20. Mean Field Games (2020)
  21. Optimal Sharing, Equilibria, and Welfare Without Risk Aversion — → 2401.03328
  22. The Equality of the Core and the Set of Equilibria in Economies With Infinitely Many Commodities and a Continuum of Agents (1973)
  23. Weak Cauchy Sequences in $ L^ 1 (E) $ (1984)
  24. Topological Vector Spaces (1999)
  25. Functional analysis, Sobolev spaces and Partial Differential Equations (2011)
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  27. Efficiency in pure-exchange economies with risk-averse monetary utilities — → 2406.02712
  28. Pareto-Efficient Risk Exchange with a Continuum of Agents (2026)
  29. Economic Analysis of Law (2014)

3 of 37 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.