Paper: arXiv 2409.05194

Authors: Vasily Melnikov

Abstract

We study risk measures $\varphi:E\longrightarrow\mathbb{R}\cup{\infty}$, where $E$ is a vector space of random variables which a priori has no lattice structure$\unicode{x2014}$a blind spot of the existing risk measures literature. In particular, we address when $\varphi$ admits a tractable dual representation (one which does not contain non-$σ$-additive signed measures), and whether one can extend $\varphi$ to a solid superspace of $E$. The existence of a tractable dual representation is shown to be equivalent, modulo certain technicalities, to a Fatou-like property, while extension theorems are established under the existence of a sufficiently regular lift, a potentially non-linear mechanism of assigning random variable extensions to certain linear functionals on $E$. Our motivation is broadening the theory of risk measures to spaces without a lattice structure, which are ubiquitous in financial economics, especially when markets are incomplete.

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 framework extending risk measure theory to non-solid vector spaces, addressing a fundamental theoretical gap for incomplete markets. It relies entirely on functional analysis and probability theory without empirical data, backtests, or code.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop Risk Measures on Non-Solid/Non-Lattice Vector Spaces] --> B[Input: Vector Space E without Lattice Structure in Incomplete Markets]
    B --> C[Method: Analyze Dual Representations & Construct Non-Linear Lift Mechanisms]
    C --> D[Computational Step: Evaluate Fatou-Like Properties & Super-Hedging Extensions]
    D --> E[Outcome 1: Equivalent Conditions for Tractable Dual Representations]
    D --> F[Outcome 2: Extension Theorems to Solid Superspaces via Regular Lifts]

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