chewi_display_1_1_5 · Samplinglib

Plain-English statement

When an elementary stochastic integral is squared and averaged, every cross term between distinct time cells vanishes.

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.

Mathematical statement

E[(sum_i H_i Delta B_i)^2] = sum_i E[(H_i Delta B_i)^2].

Intuition

At the later interval's left endpoint, the earlier weighted increment and the later coefficient are already known, while the later Brownian increment is centered and independent of that past.

Conditions

bounded left-endpoint measurable coefficients
Brownian motion relative to the same filtration
a finite time grid and terminal time

Why these conditions cannot be dropped

adaptedness prevents coefficients from seeing future increments
filtration-level independence is stronger than bare pairwise increment independence
L2 bounds justify integrating the finite square expansion

Proof route

prove each weighted increment is in L2
order two distinct grid cells
factor the cross expectation using filtration independence
use the centered future increment
integrate the finite double-sum expansion

Lean interface notes

IsBrownianMotionWithFiltration records independence from the whole past sigma-algebra
MemLp.integrable_mul justifies every cross-term integral
integral_weightedIncrement_mul_eq_zero is the reusable orthogonality leaf

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Lean statement

theorem chewi_display_1_1_5
    {Ω : Type*} {m : MeasurableSpace Ω}
    {B : ℝ≥0 → Ω → ℝ} {filtration : Filtration ℝ≥0 m} {μ : Measure Ω}
    {n : ℕ} (eta : ElementaryAdaptedProcess filtration n)
    (hB : IsBrownianMotionWithFiltration B filtration μ) (T : ℝ≥0) :
    ∫ omega, elementaryItoIntegral eta B T omega ^ 2 ∂μ =
      ∑ i, ∫ omega, weightedIncrement eta B T i omega ^ 2 ∂μ := by
  let _ : IsProbabilityMeasure μ := hB.isProbabilityMeasure
  let W : Fin n → Ω → ℝ := fun i => weightedIncrement eta B T i
  have hW : ∀ i, MemLp (W i) 2 μ := fun i => weightedIncrement_memLp_two eta hB T i
  have hpair : ∀ i j, Integrable (fun omega => W i omega * W j omega) μ := by
    intro i j
    change Integrable (W i * W j) μ
    exact (hW i).integrable_mul (hW j)
  calc
    ∫ omega, elementaryItoIntegral eta B T omega ^ 2 ∂μ =
        ∫ omega, ∑ i, ∑ j, W i omega * W j omega ∂μ := by
          apply integral_congr_ae
          filter_upwards [] with omega
          rw [elementaryItoIntegral_eq_sum_weightedIncrement]
          simp only [W, Finset.sum_mul_sum, pow_two]
    _ = ∑ i, ∑ j, ∫ omega, W i omega * W j omega ∂μ := by
      rw [integral_finsetSum]
      · apply Finset.sum_congr rfl
        intro i _
        rw [integral_finsetSum]
        intro j _
        exact hpair i j
      · intro i _
        exact integrable_finsetSum _ fun j _ => hpair i j
    _ = ∑ i, ∫ omega, W i omega ^ 2 ∂μ := by
      apply Finset.sum_congr rfl
      intro i _
      rw [Finset.sum_eq_single i]
      · simp [pow_two]
      · intro j _ hji
        exact integral_weightedIncrement_mul_eq_zero eta hB T hji.symm
      · simp
    _ = ∑ i, ∫ omega, weightedIncrement eta B T i omega ^ 2 ∂μ := rfl

/-- The probabilistic part of Chewi display (1.1.6): each diagonal term is
the coefficient's second moment times the clipped time-step length. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:211published source at 7bcd37294df1