chewi_proposition_1_1_16

Plain-English statement

Chewi, Proposition 1.1.16. Assume the filtration satisfies the usual conditions, `B` is Brownian motion with respect to that filtration, and `eta` is strongly progressive with finite pathwise square energy on every finite horizon. Then the canonically glued local Itô integral process is strongly adapted, has continuous paths, and is a local martingale. The first component is included explicitly for readers even though adaptedness is already part of `Localization.IsLocalMartingale`.

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Lean statement

theorem chewi_proposition_1_1_16
    [IsProbabilityMeasure mu]
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : GlobalLocalProgressiveL2Integrand filtration mu)
    (hB : IsBrownianMotionWithFiltration B filtration mu) :
    StronglyAdapted filtration (globalItoProcess hUsual eta hB) ∧
      (∀ omega, Continuous (fun t => globalItoProcess hUsual eta hB t omega)) ∧
      Localization.IsLocalMartingale
        (globalItoProcess hUsual eta hB) filtration mu := by
  exact ⟨globalItoProcess_stronglyAdapted hUsual eta hB,
    globalItoProcess_continuous hUsual eta hB,
    globalItoProcess_isLocalMartingale hUsual eta hB⟩

/-- **Localized Itô representation for Proposition 1.1.16.**

For every canonical dyadic localizer `tau_k`, stopping the globally glued local
Itô process at `tau_k` recovers, almost surely and at every deterministic time
inside the matching horizon, the completed Itô process of the literal source
integrand `eta_s * 1_{s ≤ tau_k}`.  This is the formal certificate that the
process in `chewi_proposition_1_1_16` is Chewi's local stochastic integral, not
an unrelated local martingale with the same localization sequence. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiProposition1_1_16.lean:51published source at 7bcd37294df1

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package random-stopping consistency, cross-horizon overlap, localized martingale coherence, and pathwise gluing into the source local-Ito continuous-local-martingale result

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