Paper: arXiv 2602.00784

Authors: Tomasz Kania

Abstract

We develop a non-standard analysis framework for coherent risk measures and their finite-sample analogues, coherent risk estimators, building on recent work of Aichele, Cialenco, Jelito, and Pitera. Coherent risk measures on $L^\infty$ are realised as standard parts of internal support functionals on Loeb probability spaces, and coherent risk estimators arise as finite-grid restrictions. Our main results are: (i) a hyperfinite robust representation theorem that yields, as finite shadows, the robust representation results for coherent risk estimators; (ii) a discrete Kusuoka representation for law-invariant coherent risk estimators as suprema of mixtures of discrete expected shortfalls on ${k/n:k=1,\ldots,n}$; (iii) uniform almost sure consistency (with an explicit rate) for canonical spectral plug-in estimators over Lipschitz spectral classes; (iv) a Kusuoka-type plug-in consistency theorem under tightness and uniform estimation assumptions; (v) bootstrap validity for spectral plug-in estimators via an NSA reformulation of the functional delta method (under standard smoothness assumptions on $F_X$); and (vi) asymptotic normality obtained through a hyperfinite central limit theorem. The hyperfinite viewpoint provides a transparent probability-to-statistics dictionary: applying a risk measure to a law corresponds to evaluating an internal functional on a hyperfinite empirical measure and taking the standard part. We include a standardd self-contained introduction to the required non-standard tools.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly sophisticated theoretical framework applying non-standard analysis and Loeb probability spaces to coherent risk measures and estimators. While mathematically rigorous and novel in its approach to connecting finite-sample statistics with continuous population theory, it is entirely theoretical without empirical backtests or market data analysis.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Develop NSA framework for coherent risk measures & estimators"] --> B["Inputs:<br>Coherent risk measures on L^∞, Loeb spaces, empirical measures"]
    B --> C["Methodology:<br>Map measures to internal support functionals on hyperfinite grids"]
    C --> D["Theoretical Formulations:<br>Hyperfinite Robust Representation & Discrete Kusuoka Formulae"]
    D --> E["Asymptotic Analysis:<br>Functional Delta Method & Hyperfinite CLT for Plug-in Estimators"]
    E --> F["Key Outcomes:<br>Uniform a.s. consistency, Bootstrap validity, & Asymptotic normality"]