integral_itoIntegralProcess_at_boundedStopping_eq_zero

Plain-English statement

- The completed Itô process has mean zero when evaluated at a stopping time that is pointwise bounded by its construction horizon.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem integral_itoIntegralProcess_at_boundedStopping_eq_zero
    [IsFiniteMeasure mu]
    (eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T)
    (tau : Omega → ℝ≥0)
    (htau : IsChewiStoppingTime filtration
      (fun omega => (tau omega : WithTop ℝ≥0)))
    (htauT : ∀ omega, tau omega ≤ T)
    (hB : IsBrownianMotionWithFiltration B filtration mu)
    (hUsual : SatisfiesUsualConditions filtration mu) :
    ∫ omega, itoIntegralProcess eta hT hB hUsual (tau omega) omega ∂mu = 0 := by
  let _ : IsProbabilityMeasure mu := hB.isProbabilityMeasure
  let stoppedEta := stop eta (fun omega => (tau omega : WithTop ℝ≥0)) htau
  have hmartingale :
      Martingale (itoIntegralProcess stoppedEta hT hB hUsual) filtration mu :=
    itoIntegralProcess_martingale stoppedEta hT hB hUsual
  have hterminalMean :
      ∫ omega, terminalRepresentative stoppedEta hT hB omega ∂mu = 0 := by
    have hmean := hmartingale.setIntegral_eq (show (0 : ℝ≥0) ≤ T from bot_le)
      (MeasurableSet.univ : MeasurableSet[filtration 0] Set.univ)
    have hprocessMean :
        ∫ omega, itoIntegralProcess stoppedEta hT hB hUsual T omega ∂mu = 0 := by
      simpa only [Measure.restrict_univ, itoIntegralProcess_at_zero,
        Pi.zero_apply, integral_zero] using hmean.symm
    exact integral_congr_ae
      (terminalRepresentative_ae_eq_actual stoppedEta hT hB hUsual) |>.trans hprocessMean
  calc
    ∫ omega, itoIntegralProcess eta hT hB hUsual (tau omega) omega ∂mu =
        ∫ omega, terminalRepresentative stoppedEta hT hB omega ∂mu := by
          apply integral_congr_ae
          exact (itoIntegralTerminal_stop_ae eta hT tau htau htauT hB hUsual).symm
    _ = 0 := hterminalMean

end BoundedStoppedItoMean
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BoundedStoppedItoMean.lean:55published source at ed709a7f320d

Proof architecture

replace an abstract optional-stopping or centered-martingale premise by the constructed completed Ito process at a bounded stopping time

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Almost-everywhere hypotheses depend on the stated measure and representative.