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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import OAI.Probability.FreeIsing.Brownian.ArrayInnovation
import OAI.Probability.FreeIsing.Stochastic.PlantedBrownianLinearCombination
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
open Problem367.ConditionalCharacteristic
namespace Problem367
theorem array_innovation_unit_projection_setIntegral_character
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
{B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ i, IsPreBrownianReal (B i) (μ i))
(hmB : ∀ i t, Measurable (B i t))
{σ : S → ι → ℝ} (hσ : Measurable σ)
{C : ℝ≥0} (hσbound : ∀ i, ∀ᵐ s ∂ν, |σ s i| ≤ (C : ℝ))
{m : ι → ℝ≥0 → S × (ι → Ω) → ℝ}
(hm : ∀ i, IsStronglyProgressive (arrayObservationFiltration B hmB σ hσ) (m i))
(hposterior : ∀ i t, m i t =ᵐ[ν.prod (Measure.infinitePi μ)]
(ν.prod (Measure.infinitePi μ))[(fun q : S × (ι → Ω) => σ q.1 i) |
arrayObservationFiltration B hmB σ hσ t])
(I : Finset ι) (a : ι → ℝ) (ha : ∑ i ∈ I, a i ^ 2 = 1)
{s t : ℝ≥0} (hst : s ≤ t) {A : Set (S × (ι → Ω))}
(hA : MeasurableSet[arrayObservationFiltration B hmB σ hσ s] A) (u : ℝ) :
(∫ q in A, character (fun q => ∑ i ∈ I, a i *
((plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q) -
(plantedArrayObservation B σ i s q - nnrealTimePrimitive (m i) s q))) u q
∂ν.prod (Measure.infinitePi μ)) =
(ν.prod (Measure.infinitePi μ)).real A *
Complex.exp (-(((t - s : ℝ≥0) : ℝ) : ℂ) * (u : ℂ)^2 / 2) := by
classical
have (i : ι) : IsProbabilityMeasure (μ i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
let P := ν.prod (Measure.infinitePi μ)
let F := arrayObservationFiltration B hmB σ hσ
let W : ℝ≥0 → S × (ι → Ω) → ℝ := fun t q => ∑ i ∈ I, a i *
(plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q)
let N : ℝ≥0 → S × (ι → Ω) → ℝ := fun t q => ∑ i ∈ I, a i * B i t (q.2 i)
have hN : IsPreBrownianReal N P :=
planted_finite_projection_preBrownian ν μ hB hmB I a ha
have hW : Martingale W F P := by
have h := BrownianLinearCombination.martingale_sum F
(fun i : I => preBrownian_array_innovation_martingale_of_progressive_posterior
ν μ hB hmB hσ i (hσbound i) (hm i) (hposterior i)) (fun i : I => a i)
convert h using 1
funext t q
exact (Finset.sum_coe_sort I (fun i => a i *
(plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q))).symm
have hvariance : ∀ r v : ℝ≥0, r ≤ v →
P[(fun q => (N v q - N r q)^2) | F r] =ᵐ[P]
fun _ => (v : ℝ) - (r : ℝ) := by
intro r v hrv
have h := planted_weighted_increment_sq_condExp_subfield ν μ hB hmB hrv I a
(arrayObservationFiltration_le_signalArrayFiltration B hmB σ hσ r)
simpa only [N, P, F, mul_sub, Finset.sum_sub_distrib, ha, mul_one] using h
have hbound : ∀ r v : ℝ≥0, r ≤ v → ∀ᵐ q ∂P,
|(W v q - W r q) - (N v q - N r q)| ≤
(2 * (C : ℝ) * ∑ i ∈ I, |a i|) * ((v : ℝ) - (r : ℝ)) := by
intro r v hrv
have hcoords : ∀ᵐ q ∂P, ∀ i : I,
|((plantedArrayObservation B σ i v q - nnrealTimePrimitive (m i) v q) -
(plantedArrayObservation B σ i r q - nnrealTimePrimitive (m i) r q)) -
(B i v (q.2 i) - B i r (q.2 i))| ≤
(2 * (C : ℝ)) * ((v : ℝ) - (r : ℝ)) :=
ae_all_iff.mpr fun i => ae_array_innovation_increment_drift_bound
ν μ hmB hσ i (hσbound i) (hm i) (hposterior i) hrv
filter_upwards [hcoords] with q hq
let D : ι → ℝ := fun i =>
((plantedArrayObservation B σ i v q - nnrealTimePrimitive (m i) v q) -
(plantedArrayObservation B σ i r q - nnrealTimePrimitive (m i) r q)) -
(B i v (q.2 i) - B i r (q.2 i))
have heq : (W v q - W r q) - (N v q - N r q) = ∑ i ∈ I, a i * D i := by
simp only [W, N, D, mul_sub, Finset.sum_sub_distrib]
rw [heq]
calc
|∑ i ∈ I, a i * D i| ≤ ∑ i ∈ I, |a i * D i| := Finset.abs_sum_le_sum_abs _ _
_ = ∑ i ∈ I, |a i| * |D i| := by simp only [abs_mul]
_ ≤ ∑ i ∈ I, |a i| * ((2 * (C : ℝ)) * ((v : ℝ) - (r : ℝ))) := by
apply Finset.sum_le_sum
intro i hi
exact mul_le_mul_of_nonneg_left (hq ⟨i, hi⟩) (abs_nonneg _)
_ = _ := by rw [← Finset.sum_mul]; ring
have h := BrownianInnovation.setIntegral_character_of_bounded_drift hN hW hvariance
(by positivity : 0 ≤ 2 * (C : ℝ) * ∑ i ∈ I, |a i|) hbound hst hA u
simpa only [W, mul_sub, Finset.sum_sub_distrib] using h
theorem array_innovation_unit_projection_condExp_character
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
{B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ i, IsPreBrownianReal (B i) (μ i))
(hmB : ∀ i t, Measurable (B i t))
{σ : S → ι → ℝ} (hσ : Measurable σ)
{C : ℝ≥0} (hσbound : ∀ i, ∀ᵐ s ∂ν, |σ s i| ≤ (C : ℝ))
{m : ι → ℝ≥0 → S × (ι → Ω) → ℝ}
(hm : ∀ i, IsStronglyProgressive (arrayObservationFiltration B hmB σ hσ) (m i))
(hposterior : ∀ i t, m i t =ᵐ[ν.prod (Measure.infinitePi μ)]
(ν.prod (Measure.infinitePi μ))[(fun q : S × (ι → Ω) => σ q.1 i) |
arrayObservationFiltration B hmB σ hσ t])
(I : Finset ι) (a : ι → ℝ) (ha : ∑ i ∈ I, a i ^ 2 = 1)
{s t : ℝ≥0} (hst : s ≤ t) (u : ℝ) :
(ν.prod (Measure.infinitePi μ))[character (fun q => ∑ i ∈ I, a i *
((plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q) -
(plantedArrayObservation B σ i s q - nnrealTimePrimitive (m i) s q))) u |
arrayObservationFiltration B hmB σ hσ s] =ᵐ[ν.prod (Measure.infinitePi μ)]
fun _ => Complex.exp (-(((t - s : ℝ≥0) : ℝ) : ℂ) * (u : ℂ)^2 / 2) := by
classical
have (i : ι) : IsProbabilityMeasure (μ i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
have hW (i : ι) (r : ℝ≥0) : Measurable (fun q =>
plantedArrayObservation B σ i r q - nnrealTimePrimitive (m i) r q) :=
((preBrownian_array_innovation_martingale_of_progressive_posterior
ν μ hB hmB hσ i (hσbound i) (hm i) (hposterior i)).stronglyMeasurable r).mono
((arrayObservationFiltration B hmB σ hσ).le r) |>.measurable
apply BrownianCharacterization.condExp_character_of_setIntegral
((arrayObservationFiltration B hmB σ hσ).le s)
· exact Finset.measurable_sum I fun i _ => ((hW i t).sub (hW i s)).const_mul (a i)
· intro A hA
exact array_innovation_unit_projection_setIntegral_character
ν μ hB hmB hσ hσbound hm hposterior I a ha hst hA u
end Problem367
end
end OAI
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