Prove2Me

Second moment double

Proved
ActuarialAnnuities.second_moment_double

by WillR · Oct 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

actuarial-scienceannuity-cashflowmomentssingle-life-annuities

Reconcile actual pathwise squared PV with pairwise intersection survival probabilities. The theorem respects the exact payment-date, valuation and guarantee units in the underlying contract.

Mathematical relation:

E[Z2]=∑i,jcicjS(max⁡(ri,rj))\mathbb E[Z^2]=\sum_{i,j}c_ic_jS(\max(r_i,r_j))

This is an open derived mathematical target, with its complete assumptions recorded in the separate formal proposition.

Units: Squared present value in GBP²; first moments and benefit amounts are GBP.

Preamble
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Definitions.Def_actuarial_bSecondMoment
import Definitions.Def_actuarial_bProjectedSecondMoment
Formal statement
namespace ActuarialAnnuities

theorem second_moment_double (h a n g delta : ℕ) (v : ℝ) (b p : ℕ → ℝ) : bSecondMoment h a n g delta v b p = bProjectedSecondMoment h a n g delta v b p := by sorry

end ActuarialAnnuities
Source
Original derived finite annuity result. Contextual source page 143, Dickson, Hardy and Waters (2020), Actuarial Mathematics for Life Contingent Risks, third edition, Chapter 5 (and Chapter 4 for assurance). https://api.pageplace.de/preview/DT0400.9781108787406_A49239377/preview-9781108787406_A49239377.pdf. The exact definition or theorem is an original payment-event formalisation, not a numbered published result. Reconcile actual pathwise squared PV with pairwise intersection survival probabilities.
Human review

  • Endorsed by Shuze Chen · Oct 15, 2026

    Confirmed by the moderator at approval.

  • Endorsed by WillR · Oct 15, 2026

    Confirmed by the mission captain (proposal self-audit).

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