Stop-loss reinsurance, retained deficit and pathwise capital: ruStopLoss_zero_below

Proved

ActuarialValuation.ruStopLoss_zero_below

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

actuarial-scienceformal-verificationruin-deficit-process

Claims below the attachment point create no reinsurer liability. The full Lean declaration specifies the finite boundaries, exact units and any positivity, independence or regularity assumptions. This is an original derived Actuarial mathematical statement, not a claim of a numbered previously published theorem.

Mathematical relation:

X≤d⇒Y=0

Preamble

import Mathlib.Data.Real.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Definitions.Def_actuarial_ruStopLoss

Formal statement

namespace ActuarialValuation

theorem ruStopLoss_zero_below (x d : ℝ) (h : x ≤ d) : ruStopLoss x d = 0 := by sorry

end ActuarialValuation

Source

Original derived actuarial mathematics, published source page 433. S. David Promislow, Fundamentals of Actuarial Mathematics, third edition (2015), Chapter 23, https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/ftoc.xhtml; Palmowski, Ramsden and Papaioannou (2018), Parisian ruin for the dual risk process in discrete-time, https://link.springer.com/article/10.1007/s13385-018-0172-8. Parent topic: Actuarial surplus and ruin: first-passage event, claim timing, nonnegative deficit, retained loss and reinsurance comparison for finite portfolios. 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://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/ftoc.xhtml

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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