Paper: arXiv 2509.08832

Authors: Vasily Melnikov

Abstract

We consider the optimal risk sharing problem with a continuum of agents, modeled via a non-atomic measure space. Individual preferences are not assumed to be convex. We show the multiplicity of agents induces the value function to be convex, allowing for the application of convex duality techniques to risk sharing without preference convexity. The proof in the finite-dimensional case is based on aggregate convexity principles emanating from Lyapunov convexity, while the infinite-dimensional case uses the finite-dimensional results conjoined with approximation arguments particular to a class of law invariant risk measures, although the reference measure is allowed to vary between agents. Finally, we derive a computationally tractable formula for the conjugate of the value function, yielding an explicit dual representation of the value function.

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 highly sophisticated theoretical framework using Lyapunov convexity, functional analysis, and Gelfand integration to solve non-convex optimal risk sharing in continuous agent spaces. However, it is purely mathematical theory without empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Optimal Risk Sharing without Convex Preferences] --> B[Inputs: Continuum of Agents & Non-Atomic Measure Space]
    B --> C[Method: Leverage Lyapunov Convexity for Aggregate Value Function]
    C --> D[Approximation: Extend Finite-Dim Results to Infinite-Dim Law-Invariant Risk Measures]
    D --> E[Computation: Apply Convex Duality & Derive Conjugate Formula]
    E --> F[Outcome: Explicit Dual Representation & Tractable Risk Sharing Model]

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3 of 63 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.