| import OAI.Probability.FreeIsing.Brownian.ArrayInnovation |
| import OAI.Probability.FreeIsing.Stochastic.PlantedBrownianLinearCombination |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
| open Problem367.ConditionalCharacteristic |
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| namespace Problem367 |
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| 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 |
|
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| 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 |
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| end Problem367 |
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| end |
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| end OAI |
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