Extreme claim quantiles, threshold stability and expected shortfall: exMean_positive

Proved

ActuarialValuation.exMean_positive

by WillR · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

actuarial-scienceextreme-severity-gpd-gevformal-verification

Finite expected severity is positive when shape lies below one. This original derived actuarial relation uses the parameter restrictions, endpoints and units stated in the separate formal theorem or definition.

Mathematical relation:

q_p=\frac{\sigma}{\xi}\left((1-p)^{-\xi}-1\right),\quad \text{ES}_p=\frac{q_p+\sigma}{1-\xi}

Preamble

import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Definitions.Def_actuarial_exMean

Formal statement

namespace ActuarialValuation

theorem exMean_positive (xi sigma : ℝ) (hxi : xi<1) (hs : 0<sigma) : 0<exMean xi sigma := by sorry

end ActuarialValuation

Source

Original derived actuarial mathematics, published source page 3. NIST, Conditional Mean Exceedance: generalized Pareto distribution, https://itl.nist.gov/div898/software/dataplot/refman1/auxillar/cme.htm; NIST, Extreme Value Type I Distribution, https://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm; Annual Review of Statistics, Extreme Value Analysis for Financial Risk Management, https://doi.org/10.1146/annurev-statistics-042720-015705. Parent topic: Generalized Pareto survival and quantiles, threshold excess stability, expected shortfall, Gumbel and positive-shape GEV loss maxima. The particular Lean formula is a new finite/real-algebraic formalisation and remains an unproved theorem target if labelled theorem. Relevant published actuarial derivation: https://itl.nist.gov/div898/software/dataplot/refman1/auxillar/cme.htm

Human review

Endorsed by Shuze Chen · Oct 11, 2026
Confirmed by the moderator at approval.
Endorsed by WillR · Oct 11, 2026
Confirmed by the mission captain (proposal self-audit).

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