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

TL;DR

This paper introduces a non-standard analysis framework for coherent risk measures, providing hyperfinite representations, discrete Kusuoka formulae, and asymptotic results for plug-in estimators, enhancing risk estimation theory.

Contribution

It develops a hyperfinite robust representation theorem and discrete Kusuoka representations, along with consistency and asymptotic normality results for risk estimators using NSA techniques.

Findings

Hyperfinite robust representation theorem established.

Discrete Kusuoka representation for law-invariant risk estimators.

Proven consistency and asymptotic normality of plug-in estimators.

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.