AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation

- Zero extension of a clipped progressive process from `[0,b] x Omega`. The target measurable space on `Omega` is the filtration at time `b`.

noncomputable def clippedExtensionAt
    (eta : ProgressiveL2Integrand filtration mu T) (truncationLevel : ℕ)
    (b : ℝ≥0) : ℝ≥0 × Omega → ℝ :=
  Function.extend
    (Prod.map ((↑) : Set.Iic b → ℝ≥0) id)
    (fun p : Set.Iic b × Omega =>
      (clipped eta truncationLevel).process p.1 p.2)
    0

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:30published source at 77184245109a

No declaration docstring.

theorem clippedExtensionAt_stronglyMeasurable
    (eta : ProgressiveL2Integrand filtration mu T) (truncationLevel : ℕ)
    (b : ℝ≥0) :
    @StronglyMeasurable (ℝ≥0 × Omega) ℝ inferInstance
      (MeasurableSpace.prod inferInstance (filtration b))
      (clippedExtensionAt eta truncationLevel b) := by
  apply ((MeasurableEmbedding.subtype_coe measurableSet_Iic).prodMap
    MeasurableEmbedding.id).stronglyMeasurable_extend
  · exact (clipped eta truncationLevel).progressive b
  · exact stronglyMeasurable_const

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:39published source at 77184245109a

No declaration docstring.

@[simp] theorem clippedExtensionAt_apply_of_le
    (eta : ProgressiveL2Integrand filtration mu T) (truncationLevel : ℕ)
    {s b : ℝ≥0} (hsb : s ≤ b) (omega : Omega) :
    clippedExtensionAt eta truncationLevel b (s, omega) =
      (clipped eta truncationLevel).process s omega := by
  let p : Set.Iic b × Omega := (⟨s, hsb⟩, omega)
  exact ((MeasurableEmbedding.subtype_coe measurableSet_Iic).prodMap
    MeasurableEmbedding.id).injective.extend_apply
      (fun q : Set.Iic b × Omega =>
        (clipped eta truncationLevel).process q.1 q.2) 0 p

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:50published source at 77184245109a

No declaration docstring.

theorem clippedExtensionAt_abs_le
    (eta : ProgressiveL2Integrand filtration mu T) (truncationLevel : ℕ)
    (b s : ℝ≥0) (omega : Omega) :
    |clippedExtensionAt eta truncationLevel b (s, omega)| ≤
      (truncationLevel : ℝ) := by
  by_cases hsb : s ≤ b
  · rw [clippedExtensionAt_apply_of_le eta truncationLevel hsb]
    exact clipped_abs_le eta truncationLevel s omega
  · rw [clippedExtensionAt, Function.extend_apply']
    · simp
    · rintro ⟨u, hu⟩
      apply hsb
      have hsu := congrArg Prod.fst hu
      change (u.1 : ℝ≥0) = s at hsu
      have hub : (u.1 : ℝ≥0) ≤ b := by
        exact Set.mem_Iic.mp u.1.property
      rwa [hsu] at hub

/-- Average of the clipped process over `(a,b]`, normalized by `delta`.
The extension makes the joint measurability used by parameterized Bochner
integration explicit. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:61published source at 77184245109a

- Average of the clipped process over `(a,b]`, normalized by `delta`. The extension makes the joint measurability used by parameterized Bochner integration explicit.

noncomputable def laggedCellAverage
    (eta : ProgressiveL2Integrand filtration mu T) (truncationLevel : ℕ)
    (delta a b : ℝ≥0) (omega : Omega) : ℝ :=
  (delta : ℝ)⁻¹ *
    ∫ s, clippedExtensionAt eta truncationLevel b (s, omega)
      ∂((TimeMeasure.upTo T).restrict (Set.Ioc a b))

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:82published source at 77184245109a

No declaration docstring.

theorem laggedCellAverage_stronglyMeasurable
    (eta : ProgressiveL2Integrand filtration mu T) (truncationLevel : ℕ)
    (delta a b : ℝ≥0) :
    @StronglyMeasurable Omega ℝ inferInstance (filtration b)
      (laggedCellAverage eta truncationLevel delta a b) := by
  have hintegral :
      @StronglyMeasurable Omega ℝ inferInstance (filtration b)
        (fun omega =>
          ∫ s, clippedExtensionAt eta truncationLevel b (s, omega)
            ∂((TimeMeasure.upTo T).restrict (Set.Ioc a b))) :=
    @StronglyMeasurable.integral_prod_left'
      ℝ≥0 Omega ℝ inferInstance (filtration b)
      ((TimeMeasure.upTo T).restrict (Set.Ioc a b)) inferInstance inferInstance inferInstance
      (clippedExtensionAt eta truncationLevel b)
      (clippedExtensionAt_stronglyMeasurable eta truncationLevel b)
  exact stronglyMeasurable_const.mul hintegral

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:89published source at 77184245109a

No declaration docstring.

theorem laggedCellAverage_zero
    (eta : ProgressiveL2Integrand filtration mu T) (truncationLevel : ℕ)
    (delta a : ℝ≥0) :
    laggedCellAverage eta truncationLevel delta a a = 0 := by
  funext omega
  simp [laggedCellAverage]

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:106published source at 77184245109a

No declaration docstring.

theorem laggedCellAverage_abs_le
    (eta : ProgressiveL2Integrand filtration mu T) (truncationLevel : ℕ)
    {delta a b : ℝ≥0} (hdelta : 0 < delta) (hab : a ≤ b)
    (hbT : b ≤ T) (hcell : b - a = delta) (omega : Omega) :
    |laggedCellAverage eta truncationLevel delta a b omega| ≤
      (truncationLevel : ℝ) := by
  have hmeasure :
      ((TimeMeasure.upTo T).restrict (Set.Ioc a b)).real Set.univ =
        (delta : ℝ) := by
    rw [Measure.real, Measure.restrict_apply MeasurableSet.univ]
    simp only [Set.univ_inter]
    rw [TimeMeasure.upTo_Ioc T a b hab]
    rw [min_eq_left hbT, min_eq_left (hab.trans hbT), hcell]
    simp
  have hintegral :
      |∫ s, clippedExtensionAt eta truncationLevel b (s, omega)
          ∂((TimeMeasure.upTo T).restrict (Set.Ioc a b))| ≤
        (truncationLevel : ℝ) * (delta : ℝ) := by
    have hbound := norm_integral_le_of_norm_le_const
      (μ := (TimeMeasure.upTo T).restrict (Set.Ioc a b))
      (f := fun s => clippedExtensionAt eta truncationLevel b (s, omega))
      (Filter.Eventually.of_forall fun s => by
        simpa only [Real.norm_eq_abs] using
          clippedExtensionAt_abs_le eta truncationLevel b s omega)
    simpa only [Real.norm_eq_abs, hmeasure] using hbound
  have hdeltaReal : 0 < (delta : ℝ) := by exact_mod_cast hdelta
  rw [laggedCellAverage, abs_mul, abs_inv, abs_of_pos hdeltaReal]
  calc
    (delta : ℝ)⁻¹ *
        |∫ s, clippedExtensionAt eta truncationLevel b (s, omega)
          ∂((TimeMeasure.upTo T).restrict (Set.Ioc a b))| ≤
        (delta : ℝ)⁻¹ * ((truncationLevel : ℝ) * (delta : ℝ)) :=
      mul_le_mul_of_nonneg_left hintegral (inv_nonneg.mpr hdeltaReal.le)
    _ = (truncationLevel : ℝ) := by
      field_simp

/-- The left endpoint of the dyadic cell indexed by `i`. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:113published source at 77184245109a

- The left endpoint of the dyadic cell indexed by `i`.

noncomputable def dyadicLeftTime (T : ℝ≥0) (level : ℕ) (i : ℕ) : ℝ≥0 :=
  (i : ℝ≥0) * dyadicMesh T level

/-- The lagged coefficient used on dyadic cell `i`.  Cell zero has coefficient
zero; every later cell uses the average over the immediately preceding cell. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:150published source at 77184245109a

- The lagged coefficient used on dyadic cell `i`. Cell zero has coefficient zero; every later cell uses the average over the immediately preceding cell.

noncomputable def laggedDyadicCoeff
    (eta : ProgressiveL2Integrand filtration mu T) (level truncationLevel : ℕ)
    (i : Fin (2 ^ level)) (omega : Omega) : ℝ :=
  if _hi : i.val = 0 then 0 else
    laggedCellAverage eta truncationLevel (dyadicMesh T level)
      (dyadicLeftTime T level i.val - dyadicMesh T level)
      (dyadicLeftTime T level i.val) omega

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:155published source at 77184245109a

No declaration docstring.

theorem dyadicMesh_le_leftTime_of_ne_zero
    {T : ℝ≥0} (level : ℕ) {i : ℕ} (hi : i ≠ 0) :
    dyadicMesh T level ≤ dyadicLeftTime T level i := by
  have hiOne : 1 ≤ i := Nat.one_le_iff_ne_zero.mpr hi
  have hiOneNN : (1 : ℝ≥0) ≤ (i : ℝ≥0) := by exact_mod_cast hiOne
  simpa only [dyadicLeftTime, one_mul] using
    mul_le_mul_of_nonneg_right hiOneNN (by positivity)

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:163published source at 77184245109a

No declaration docstring.

private theorem dyadicLeftTime_le_terminal
    (eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T)
    (level truncationLevel : ℕ) (i : Fin (2 ^ level)) :
    dyadicLeftTime T level i.val ≤ T := by
  have hmono : Monotone (regularGridTimes (dyadicMesh T level) (2 ^ level)) :=
    (regularGridTimes_strictMono (dyadicMesh_pos hT level) (2 ^ level)).monotone
  have hle := hmono (Fin.le_last i.castSucc)
  have hlast := sampledClippedDyadic_last_time eta hT level truncationLevel
  rw [sampledClippedDyadic, sampledClipped_times] at hlast
  change dyadicLeftTime T level i.val ≤
    regularGridTimes (dyadicMesh T level) (2 ^ level) (Fin.last (2 ^ level)) at hle
  exact hle.trans_eq hlast

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:171published source at 77184245109a

No declaration docstring.

theorem laggedDyadicCoeff_stronglyMeasurable
    (eta : ProgressiveL2Integrand filtration mu T) (level truncationLevel : ℕ)
    (i : Fin (2 ^ level)) :
    StronglyMeasurable[filtration (dyadicLeftTime T level i.val)]
      (laggedDyadicCoeff eta level truncationLevel i) := by
  by_cases hi : i.val = 0
  · have hzero :
        @StronglyMeasurable Omega ℝ inferInstance
          (filtration (dyadicLeftTime T level i.val)) (fun _ => 0) :=
      stronglyMeasurable_const
    convert hzero using 1
    funext omega
    simp [laggedDyadicCoeff, hi]
  · convert laggedCellAverage_stronglyMeasurable eta truncationLevel
      (dyadicMesh T level)
      (dyadicLeftTime T level i.val - dyadicMesh T level)
      (dyadicLeftTime T level i.val) using 1
    funext omega
    simp [laggedDyadicCoeff, hi]

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:184published source at 77184245109a

No declaration docstring.

theorem laggedDyadicCoeff_abs_le
    (eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T)
    (level truncationLevel : ℕ) (i : Fin (2 ^ level)) (omega : Omega) :
    |laggedDyadicCoeff eta level truncationLevel i omega| ≤
      (truncationLevel : ℝ) := by
  by_cases hi : i.val = 0
  · simp [laggedDyadicCoeff, hi]
  · simp only [laggedDyadicCoeff, hi, ↓reduceDIte]
    let delta := dyadicMesh T level
    let b := dyadicLeftTime T level i.val
    have hdelta : 0 < delta := dyadicMesh_pos hT level
    have hdelta_b : delta ≤ b :=
      dyadicMesh_le_leftTime_of_ne_zero level hi
    have hcell : b - (b - delta) = delta :=
      tsub_tsub_cancel_of_le hdelta_b
    exact laggedCellAverage_abs_le eta truncationLevel hdelta
      (tsub_le_self) (dyadicLeftTime_le_terminal eta hT level truncationLevel i)
      hcell omega

/-- The bounded elementary adapted process obtained by lagging dyadic cell
averages by one cell. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:204published source at 77184245109a

- The bounded elementary adapted process obtained by lagging dyadic cell averages by one cell.

noncomputable def laggedDyadicApprox
    (eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T)
    (level truncationLevel : ℕ) :
    ElementaryAdaptedProcess filtration (2 ^ level) where
  times := regularGridTimes (dyadicMesh T level) (2 ^ level)
  times_strictMono := regularGridTimes_strictMono (dyadicMesh_pos hT level) _
  coeff := laggedDyadicCoeff eta level truncationLevel
  coeff_stronglyMeasurable := fun i => by
    change StronglyMeasurable[filtration (dyadicLeftTime T level i.val)]
      (laggedDyadicCoeff eta level truncationLevel i)
    exact laggedDyadicCoeff_stronglyMeasurable eta level truncationLevel i
  coeff_bounded := fun i =>
    ⟨truncationLevel, laggedDyadicCoeff_abs_le eta hT level truncationLevel i⟩

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:225published source at 77184245109a

No declaration docstring.

@[simp] theorem laggedDyadicApprox_times
    (eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T)
    (level truncationLevel : ℕ) :
    (laggedDyadicApprox eta hT level truncationLevel).times =
      regularGridTimes (dyadicMesh T level) (2 ^ level) :=
  rfl

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:239published source at 77184245109a

No declaration docstring.

@[simp] theorem laggedDyadicApprox_coeff
    (eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T)
    (level truncationLevel : ℕ) (i : Fin (2 ^ level)) (omega : Omega) :
    (laggedDyadicApprox eta hT level truncationLevel).coeff i omega =
      laggedDyadicCoeff eta level truncationLevel i omega :=
  rfl

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:246published source at 77184245109a

No declaration docstring.

theorem laggedDyadicApprox_last_time
    (eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T)
    (level truncationLevel : ℕ) :
    (laggedDyadicApprox eta hT level truncationLevel).times
        (Fin.last (2 ^ level)) = T := by
  have hlast := sampledClippedDyadic_last_time eta hT level truncationLevel
  simpa only [sampledClippedDyadic, sampledClipped_times,
    laggedDyadicApprox_times] using hlast

/-- The two obligations that matter downstream: each coefficient is known at
its cell's left endpoint and remains bounded by the clipping level. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:253published source at 77184245109a

- The two obligations that matter downstream: each coefficient is known at its cell's left endpoint and remains bounded by the clipping level.

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

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:264published source at 77184245109aOpen detailed card