Non-standard analysis for coherent risk estimation: hyperfinite representations, discrete Kusuoka formulae, and plug-in asymptotics
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911494888751104 |
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| author | Kania, Tomasz |
| author_facet | Kania, Tomasz |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_00784 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-standard analysis for coherent risk estimation: hyperfinite representations, discrete Kusuoka formulae, and plug-in asymptotics Kania, Tomasz Risk Management Logic Probability Statistics Theory Mathematical Finance 60B10, 60F15, 62G30, 91G70, 03H05 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. |
| title | Non-standard analysis for coherent risk estimation: hyperfinite representations, discrete Kusuoka formulae, and plug-in asymptotics |
| topic | Risk Management Logic Probability Statistics Theory Mathematical Finance 60B10, 60F15, 62G30, 91G70, 03H05 |
| url | https://arxiv.org/abs/2602.00784 |