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

Tomasz Kania|arXiv (Cornell University)|Jan 31, 2026

Risk and Portfolio Optimization0 citations

TL;DR

The paper develops a non-standard analysis framework to unify coherent risk measures and their finite-sample estimators, providing hyperfinite representations, discrete Kusuoka representations, and plug-in asymptotics for spectral CREs.

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.

Motivation & Objective

• Motivate the estimation of coherent risk measures (CRMs) from finite samples and bridge population and sample theories.
• Provide a unified NSA framework where CRMs on L^ ∞ are standard parts of internal functionals on Loeb spaces, and CREs are finite shadows.
• Derive finite-sample robust representations and new representations for law-invariant CREs, including discrete Kusuoka representations.
• Establish consistency, bootstrap validity, and asymptotic normality results for spectral plug-in estimators under NSA.
• Explore extensions to Orlicz spaces and practical implications for risk estimation under model uncertainty.

Proposed method

• Construct a hyperfinite representation of CRMs as standard parts of internal functionals on Loeb probability spaces.
• Show that CREs arise as finite shadows of CRMs via discrete grid restrictions.
• Derive a discrete Kusuoka representation for law-invariant CREs as suprema of discrete expected shortfalls on grid points {k/n}.
• Prove uniform almost sure consistency for spectral plug-in estimators over Lipschitz spectral classes with explicit rates.
• Provide a Kusuoka-type plug-in consistency theorem under tightness and uniform estimation assumptions.
• Establish bootstrap validity for spectral plug-in estimators using an NSA reformulation of the functional delta method.
• Obtain asymptotic normality via a hyperfinite central limit theorem.

Experimental results

Research questions

• RQ1 How can coherent risk measures on L∞ be represented as standard parts of hyperfinite functionals on Loeb spaces?
• RQ2 What are the finite-sample robust representations for coherent risk estimators and law-invariant CREs?
• RQ3 Can law-invariant CREs be represented discretely as mixtures of discrete expected shortfalls on a grid?
• RQ4 Do spectral plug-in estimators exhibit uniform consistency and explicit convergence rates across Lipschitz spectral classes?
• RQ5 Is bootstrap valid for spectral plug-in estimators under the NSA-augmented delta method, and can NSA yield asymptotic normality via a hyperfinite CLT?

Key findings

• Coherent risk measures on L∞ are the standard parts of hyperfinite internal functionals, unifying population and finite-sample viewpoints.
• Coherent risk estimators admit a hyperfinite robust representation as suprema of linear functionals over a finite simplex.
• Every law-invariant CRE has a discrete Kusuoka representation as a supremum of mixtures of discrete ES on grid points.
• Uniform almost sure consistency for canonical spectral plug-in estimators is established over Lipschitz spectral classes with an explicit rate.
• A Kusuoka-type plug-in consistency theorem is proved under tightness and uniform estimation assumptions.
• Bootstrap validity for spectral plug-in estimators is shown via an NSA reformulation of the functional delta method, under standard smoothness assumptions.
• Asymptotic normality is derived through a hyperfinite central limit theorem.