chewi_display_1_1_6 · Samplinglib

Plain-English statement

The expected squared elementary Ito integral is exactly the probability-time L2 energy of its integrand.

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[|I_[0,T](eta)|^2] = E[integral_0^T |eta_t|^2 dt].

Intuition

Each Brownian increment contributes its elapsed time as variance, so the diagonal sum is the integral of the step process's squared coefficients over its clipped time cells.

Conditions

an elementary adapted process
Brownian motion relative to the filtration
the stopped nonnegative-time Lebesgue measure

Why these conditions cannot be dropped

the Brownian law supplies increment variance
adaptedness supplies coefficient-increment independence
the time measure turns each half-open cell into its clipped length
measurability permits Tonelli on probability times time

Proof route

apply the orthogonal expansion in display (1.1.5)
evaluate every diagonal term
evaluate the process L2 energy cell by cell
identify the two finite sums

Lean interface notes

the left side is converted with ENNReal.ofReal
processL2Energy uses the product of mu and TimeMeasure.upTo T
this theorem covers elementary processes; L2 completion for the general Ito integral remains open

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Lean statement

theorem chewi_display_1_1_6
    {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) :
    ENNReal.ofReal (∫ omega, elementaryItoIntegral eta B T omega ^ 2 ∂μ) =
      processL2Energy eta.value μ T := by
  let _ : IsProbabilityMeasure μ := hB.isProbabilityMeasure
  rw [elementaryItoIntegral_sq_eq_sum eta hB T]
  rw [processL2Energy_value eta μ T]

end ElementaryItoIsometry
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:426published source at 7bcd37294df1