chewi_display_1_1_14 · Samplinglib

Plain-English statement

- Chewi display (1.1.14): every canonical energy truncation is fed into the already-constructed global Itô map and yields an adapted continuous martingale, with the exact deterministic-time restriction compatibility.

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 chewi_display_1_1_14
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : LocalProgressiveL2Integrand filtration mu T)
    (hT : 0 < T)
    (hB : IsBrownianMotionWithFiltration B filtration mu)
    (n : ℕ) :
    StronglyAdapted filtration
        (canonicalStoppedItoProcess hUsual eta hT hB n) ∧
      Martingale (canonicalStoppedItoProcess hUsual eta hT hB n)
        filtration mu ∧
      (∀ omega,
        ContinuousOn
          (fun t => canonicalStoppedItoProcess hUsual eta hT hB n t omega)
          (Icc (0 : ℝ≥0) T)) ∧
      (canonicalStoppedProgressiveL2 hUsual eta n).process =
        energyStoppedIntegrand hUsual eta (n + 1 : ℝ) ∧
      ∀ t : ℝ≥0, t ≤ T →
        canonicalStoppedItoProcess hUsual eta hT hB n t =ᵐ[mu]
          (fun omega =>
            ItoTerminalCompletion.itoIntegralTerminal
              ((canonicalStoppedProgressiveL2 hUsual eta n).restrictAt t)
              hT hB omega) := by
  refine ⟨canonicalStoppedItoProcess_stronglyAdapted hUsual eta hT hB n,
    canonicalStoppedItoProcess_martingale hUsual eta hT hB n,
    canonicalStoppedItoProcess_continuousOn hUsual eta hT hB n,
    canonicalStoppedProgressiveL2_process hUsual eta n, ?_⟩
  intro t htT
  exact canonicalStoppedItoProcess_at_eq_terminal hUsual eta hT hB n htT

end CanonicalStoppedItoIntegral
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalStoppedItoIntegral.lean:98published source at 7bcd37294df1

Proof architecture

package each canonical energy truncation as a globally L2 Ito integrand whose Ito process is adapted, continuous, martingale, and compatible with deterministic-time restriction

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