laggedDyadicApprox_isElementaryAdapted

Plain-English statement

A clipped progressive process can be replaced on a dyadic grid by bounded elementary coefficients obtained from the preceding time cell, so every coefficient is known before the cell on which it is used.

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

For dyadic mesh delta and i > 0, H_i = delta^(-1) integral_(t_(i-1),t_i] clip_M(eta_s) ds is F_(t_i)-measurable and |H_i| <= M; H_0 = 0.

Intuition

A one-cell delay makes time averaging causal. Progressive measurability on the past rectangle survives integration over time and leaves a random variable measurable at the next cell's left endpoint.

Conditions

a progressive product-space L2 integrand
positive terminal time
a natural clipping level
the finite nonnegative-time Lebesgue measure

Why these conditions cannot be dropped

progressiveness supplies joint measurability with the left-endpoint filtration
clipping supplies a uniform coefficient bound
positive terminal time gives a strict dyadic grid
the lag prevents use of future values

Proof route

clip the progressive process
zero-extend its restriction from the past time rectangle
apply parameterized Bochner integral measurability
restrict the integral to the preceding dyadic cell
bound the normalized integral by the cell mass
assemble the strict-grid elementary adapted process

Lean interface notes

StronglyMeasurable.integral_prod_left' performs the parameterized Bochner step
TimeMeasure.upTo_Ioc computes the exact cell mass
L2 convergence is certified separately by tendsto_laggedDyadicApprox_toLp_clipped

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Lean statement

theorem laggedDyadicApprox_isElementaryAdapted
    (eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T)
    (level truncationLevel : ℕ) :
    (∀ i, StronglyMeasurable[
        filtration ((laggedDyadicApprox eta hT level truncationLevel).times i.castSucc)]
        ((laggedDyadicApprox eta hT level truncationLevel).coeff i)) ∧
      (∀ i omega,
        |(laggedDyadicApprox eta hT level truncationLevel).coeff i omega| ≤
          (truncationLevel : ℝ)) := by
  constructor
  · exact (laggedDyadicApprox eta hT level truncationLevel).coeff_stronglyMeasurable
  · exact laggedDyadicCoeff_abs_le eta hT level truncationLevel

end LaggedDyadicApproximation
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:264published source at 7bcd37294df1