Prove2Me

Path Expectation Reconciliation

Proved
ActuarialAnnuities.path_expectation_reconciliation

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

actuarial-scienceannuity-cashflowexpectationsingle-life-annuities

Finite reversal of actual lifetime-state and payment-index sums derives survival projection.

Mathematical relation:

∑k=0hpkZk=∑jbjvj+δPr⁡(payment at j+δ)\sum_{k=0}^{h}p_kZ_k=\sum_jb_jv^{j+\delta}\Pr(\text{payment at }j+\delta)

Timing and admissibility conditions are the exact conditions of the declaration. Cash amounts are per instalment and present values are at valuation time zero.

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

theorem path_expectation_reconciliation (h a n g delta : ℕ) (v : ℝ) (b p : ℕ → ℝ) : bExpectedPV h a n g delta v b p = bSurvivalPV h a n g delta v b p := by sorry

end ActuarialAnnuities
Source
Original derived finite-cashflow result, contextual source page 161, Increasing annuities (Chapter 5, section 5.10). D. C. M. Dickson, M. R. Hardy and H. R. Waters (2020), Actuarial Mathematics for Life Contingent Risks, third edition, Cambridge University Press. https://www.cambridge.org/highereducation/books/actuarial-mathematics-for-life-contingent-risks/281DA4E8D523A6B23280ADC3D165AFDA/contents/4EB4E907782E2CD9CF0447D8F5DF1F6E. The cited page identifies the surrounding topic, not a numbered theorem with this exact Lean signature.
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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