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Non-standard analysis for coherent risk estimation: hyperfinite representations, discrete Kusuoka formulae, and plug-in asymptotics

Tomasz Kania

TL;DR

This work builds a unified non-standard analysis framework for coherent risk assessment, connecting population risk measures to finite-sample estimators through Loeb spaces and standard parts. It introduces a hyperfinite representation of CRMs, proves a discrete Kusuoka representation for law-invariant CREs, and establishes uniform spectral plug-in consistency, Kusuoka-type plug-in consistency, bootstrap validity, and hyperfinite CLT-based asymptotics. The approach yields a transparent probability-to-statistics dictionary and enables finite-grid approximations that preserve coherence, with extensions to Orlicz hearts. The results have practical impact for robust risk estimation from finite samples, offering rigorous discretization schemes, uniform convergence guarantees, and a solid bootstrap theory in a probabilistic-NSA setting.

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.

Non-standard analysis for coherent risk estimation: hyperfinite representations, discrete Kusuoka formulae, and plug-in asymptotics

TL;DR

This work builds a unified non-standard analysis framework for coherent risk assessment, connecting population risk measures to finite-sample estimators through Loeb spaces and standard parts. It introduces a hyperfinite representation of CRMs, proves a discrete Kusuoka representation for law-invariant CREs, and establishes uniform spectral plug-in consistency, Kusuoka-type plug-in consistency, bootstrap validity, and hyperfinite CLT-based asymptotics. The approach yields a transparent probability-to-statistics dictionary and enables finite-grid approximations that preserve coherence, with extensions to Orlicz hearts. The results have practical impact for robust risk estimation from finite samples, offering rigorous discretization schemes, uniform convergence guarantees, and a solid bootstrap theory in a probabilistic-NSA setting.

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 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 ; (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 ); 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.
Paper Structure (49 sections, 40 theorems, 182 equations)

This paper contains 49 sections, 40 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. The non-standard analysis perspective
  4. Main contributions
  5. Organisation of the paper
  6. Notation
  7. Coherent risk measures and coherent risk estimators
  8. Coherent risk measures on $L^\infty$
  9. Law invariance and spectral representation
  10. Coherent risk estimators
  11. Non-standard analysis: a self-contained introduction
  12. Standing conventions and the Loeb model
  13. The hyperreal numbers
  14. The hypernaturals and hyperfinite sets
  15. Internal and external sets
  16. ...and 34 more sections

Key Result

Theorem 2.2

Let $\rho: L^\infty \to \mathbb{R}$ be a coherent risk measure satisfying the Fatou property: if $(X_n)$ is a bounded sequence converging $\mathsf{P}$-almost surely to $X$, then $\rho(X) \leqslant \liminf_{n \to \infty} \rho(X_n)$. Then there exists a non-empty convex set $\mathcal{Q}$ of probabilit If, in addition, the set of Radon--Nikodym derivatives $\{dQ/d\mathsf{P} : Q \in \mathcal{Q}\} \sub

Theorems & Definitions (104)

  • Definition 2.1: Coherent risk measure
  • Theorem 2.2: Robust representation on $L^\infty$
  • Definition 2.3: Law invariance
  • Remark 2.4: Parameterisation convention
  • Theorem 2.5: Kusuoka representation Kusuoka
  • Definition 2.6: Spectral risk measure
  • Definition 2.7: Coherent risk estimator ACJP
  • Definition 2.8: Law invariance and comonotonicity for CREs
  • Theorem 2.9: Robust representation of CREs ACJP
  • Theorem 2.10: Law-invariant CREs ACJP
  • ...and 94 more