Prove2Me

Fractional random expectation

Proved
ActuarialAnnuities.fractional_random_expectation

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

actuarial-scienceannuity-cashflowfractionalsingle-life-annuities

Actual micro-state cashflow expectation equals an independently projected fractional UDD sum. The theorem respects the exact payment-date, valuation and guarantee units in the underlying contract.

Mathematical relation:

E[Z(m)]=∑qcqwq+δS(m)(rq)\mathbb E[Z^{(m)}]=\sum_qc_qw^{q+\delta}S^{(m)}(r_q)

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

Units: Present value in GBP per unit nominal pension, with individual instalments in GBP, payment time in anniversaries and survival as dimensionless probability.

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

theorem fractional_random_expectation (h a n g delta m : ℕ) (w : ℝ) (b p : ℕ → ℝ) : bFractionalExpectedPV h a n g delta m w b p = bFractionalSurvivalPV h a n g delta m w b p := by sorry

end ActuarialAnnuities
Source
Original derived finite annuity result. Contextual source page 152, 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. Actual micro-state cashflow expectation equals an independently projected fractional UDD sum.
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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