integral_increment_sq · Samplinglib

Plain-English statement

- The second moment of a Brownian increment is its elapsed time.

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 integral_increment_sq
    (hB : IsBrownianMotionWithFiltration B filtration μ)
    {s t : ℝ≥0} (hst : s ≤ t) :
    ∫ omega, (B t omega - B s omega) ^ 2 ∂μ = ((t - s : ℝ≥0) : ℝ) := by
  let _ : IsProbabilityMeasure μ := hB.isProbabilityMeasure
  have hpre := hB.isBrownian.toIsPreBrownianReal
  have ht : MemLp (B t) 2 μ := hpre.isGaussianProcess.hasGaussianLaw_eval t |>.memLp_two
  have hs : MemLp (B s) 2 μ := hpre.isGaussianProcess.hasGaussianLaw_eval s |>.memLp_two
  have hmean : ∫ omega, (B t omega - B s omega) ∂μ = 0 :=
    hB.integral_increment_eq_zero s t
  have hvarT : Var[B t; μ] = (t : ℝ) := by
    rw [← covariance_self ht.aemeasurable, hpre.covariance_eval, min_self]
  have hvarS : Var[B s; μ] = (s : ℝ) := by
    rw [← covariance_self hs.aemeasurable, hpre.covariance_eval, min_self]
  have hcov : cov[B t, B s; μ] = (s : ℝ) := by
    rw [hpre.covariance_eval, min_eq_right hst]
  have hvar : Var[fun omega => B t omega - B s omega; μ] =
      ((t - s : ℝ≥0) : ℝ) := by
    rw [variance_fun_sub ht hs, hvarT, hvarS, hcov, NNReal.coe_sub hst]
    ring
  have hvarianceEq := variance_eq_sub (ht.sub hs)
  have hmean' : ∫ omega, (B t - B s) omega ∂μ = 0 := by
    simpa using hmean
  rw [hmean'] at hvarianceEq
  norm_num at hvarianceEq
  have hsquare : ∫ omega, ((B t - B s) ^ 2) omega ∂μ =
      ((t - s : ℝ≥0) : ℝ) := hvarianceEq.symm.trans hvar
  simpa using hsquare

/-- The conditional second moment of a future Brownian increment is its
elapsed time. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:130published source at 7bcd37294df1

Proof architecture

replace an increment-square expectation by its elapsed time

Lean proof walkthrough

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Why the statement has this shape

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Hidden assumptions and non-claims

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