| import OAI.Probability.FreeIsing.Estimates.ArrayInnovationProjection |
| import OAI.Probability.FreeIsing.Gaussian.ArrayCharacterization |
| import OAI.Probability.FreeIsing.Stochastic.WienerSampler |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
| open Problem367.ConditionalCharacteristic |
|
|
| namespace Problem367.ArrayInnovationLaw |
|
|
| variable {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]) |
|
|
| include hB hmB hσ hσbound hm hposterior |
|
|
| theorem conditional_weighted_character |
| (I : Finset ι) (s t : ℝ≥0) (hst : s ≤ t) (a : I → 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))) 1 | |
| arrayObservationFiltration B hmB σ hσ s] =ᵐ[ν.prod (Measure.infinitePi μ)] |
| fun _ => Complex.exp (-(∑ i : I, (((t - s : ℝ≥0) : ℝ) : ℂ) * (a i : ℂ)^2) / 2) := by |
| classical |
| have (i : ι) : IsProbabilityMeasure (μ i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| apply ConditionalGaussianVector.condExp_weighted_character_of_unit_character |
| ((arrayObservationFiltration B hmB σ hσ).le s) (t - s) _ a |
| intro b hb u |
| let c : ι → ℝ := fun i => if hi : i ∈ I then b ⟨i, hi⟩ else 0 |
| have hc (i : I) : c i = b i := by simp [c, i.property] |
| have hcs : ∑ i ∈ I, c i ^ 2 = 1 := by |
| rw [← Finset.sum_coe_sort I] |
| simpa only [hc] using hb |
| have h := array_innovation_unit_projection_condExp_character |
| ν μ hB hmB hσ hσbound hm hposterior I c hcs hst u |
| have heq : (fun q => ∑ i ∈ I, c i * |
| ((plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q) - |
| (plantedArrayObservation B σ i s q - nnrealTimePrimitive (m i) s q))) = |
| (fun q => ∑ i : I, b i * |
| ((plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q) - |
| (plantedArrayObservation B σ i s q - nnrealTimePrimitive (m i) s q))) := by |
| funext q |
| rw [← Finset.sum_coe_sort I] |
| simp only [hc] |
| rw [heq] at h |
| exact h |
|
|
| theorem isGaussianProcess_innovation_array : |
| IsGaussianProcess (fun p : (_ : ι) × ℝ≥0 => fun q => |
| plantedArrayObservation B σ p.1 p.2 q - nnrealTimePrimitive (m p.1) p.2 q) |
| (ν.prod (Measure.infinitePi μ)) := by |
| classical |
| have (i : ι) : IsProbabilityMeasure (μ i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| apply GaussianArrayCharacterization.isGaussianProcess_array_of_conditional_finite_characters |
| (B := fun i t q => plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q) |
| (P := ν.prod (Measure.infinitePi μ)) (arrayObservationFiltration B hmB σ hσ) |
| · exact fun i => (preBrownian_array_innovation_martingale_of_progressive_posterior |
| ν μ hB hmB hσ i (hσbound i) (hm i) (hposterior i)).stronglyAdapted.adapted |
| · exact fun i => (preBrownian_array_innovation_of_progressive_posterior |
| ν μ hB hmB hσ i (hσbound i) (hm i) (hposterior i)).eval_zero_ae_eq_zero |
| · exact conditional_weighted_character ν μ hB hmB hσ hσbound hm hposterior |
|
|
| theorem map_innovation_array_eq_infinitePi_reference |
| {Ω' : Type*} [MeasurableSpace Ω'] {Q : Measure Ω'} |
| {R : ℝ≥0 → Ω' → ℝ} (hR : IsPreBrownianReal R Q) |
| (hmR : ∀ t, Measurable (R t)) : |
| (ν.prod (Measure.infinitePi μ)).map (fun q i t => |
| plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q) = |
| Measure.infinitePi (fun _ : ι => Q.map (fun ω t => R t ω)) := by |
| classical |
| have (i : ι) : IsProbabilityMeasure (μ i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| apply GaussianArrayCharacterization.map_brownianArray_eq_infinitePi_of_conditional_finite_characters |
| (arrayObservationFiltration B hmB σ hσ) _ _ _ hR hmR |
| · exact fun i => (preBrownian_array_innovation_martingale_of_progressive_posterior |
| ν μ hB hmB hσ i (hσbound i) (hm i) (hposterior i)).stronglyAdapted.adapted |
| · exact fun i => (preBrownian_array_innovation_of_progressive_posterior |
| ν μ hB hmB hσ i (hσbound i) (hm i) (hposterior i)).eval_zero_ae_eq_zero |
| · exact conditional_weighted_character ν μ hB hmB hσ hσbound hm hposterior |
|
|
| theorem exists_continuous_innovation_array_wiener_law |
| (hcontinuous : ∀ i, ∀ᵐ ω ∂μ i, Continuous (fun t => B i t ω)) : |
| ∃ X : ι → S × (ι → Ω) → WienerSampler.Path, |
| (∀ i, Measurable (X i)) ∧ |
| (∀ i, ∀ᵐ q ∂ν.prod (Measure.infinitePi μ), ∀ t, |
| X i q t = plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q) ∧ |
| (ν.prod (Measure.infinitePi μ)).map (fun q i => X i q) = |
| Measure.infinitePi (fun _ : ι => WienerSampler.wienerMeasure) := 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 : ι → ℝ≥0 → S × (ι → Ω) → ℝ := fun i t q => |
| plantedArrayObservation B σ i t q - nnrealTimePrimitive (m i) t q |
| have hW (i : ι) : Martingale (W i) F P := |
| preBrownian_array_innovation_martingale_of_progressive_posterior |
| ν μ hB hmB hσ i (hσbound i) (hm i) (hposterior i) |
| have hpre (i : ι) : IsPreBrownianReal (W i) P := |
| preBrownian_array_innovation_of_progressive_posterior |
| ν μ hB hmB hσ i (hσbound i) (hm i) (hposterior i) |
| have hcont (i : ι) : ∀ᵐ q ∂P, Continuous (fun t => W i t q) := |
| (brownian_array_innovation_of_progressive_posterior |
| ν μ (fun j => ⟨hB j, hcontinuous j⟩) hmB hσ i (hσbound i) |
| (hm i) (hposterior i)).cont |
| choose X hXm hXe using fun i => WienerSampler.exists_measurable_path_version |
| P (W i) (fun t => ((hW i).stronglyMeasurable t).mono (F.le t) |>.measurable) (hcont i) |
| refine ⟨X, hXm, hXe, ?_⟩ |
| have hcross := GaussianArrayCharacterization.crossCovariance_of_conditional_finite_characters |
| F (fun i => (hW i).stronglyAdapted.adapted) |
| (fun i => (hpre i).eval_zero_ae_eq_zero) |
| (conditional_weighted_character ν μ hB hmB hσ hσbound hm hposterior) |
| have hG : IsGaussianProcess (fun p : (_ : ι) × ℝ≥0 => W p.1 p.2) P := |
| isGaussianProcess_innovation_array ν μ hB hmB hσ hσbound hm hposterior |
| have hi : iIndepFun (fun i q t => X i q t) P := |
| (iIndepFun_of_gaussianArray_crossCovariance hG hcross).congr (fun i => by |
| filter_upwards [hXe i] with q hq |
| exact funext (fun t => (hq t).symm)) |
| have hiX : iIndepFun X P := iIndepFun_continuousPaths_of_evaluations hi |
| rw [hiX.map_fun_eq_infinitePi_map hXm] |
| congr 1 |
| funext i |
| have hpx : IsPreBrownianReal (fun t q => X i q t) P := (hpre i).congr (fun t => by |
| filter_upwards [hXe i] with q hq |
| exact (hq t).symm) |
| simpa only [Measure.map_id] using map_continuousPreBrownian_eq (hXm i) |
| (measurable_id : Measurable (id : WienerSampler.Path → WienerSampler.Path)) hpx |
| WienerSampler.coordinate_isBrownianReal.toIsPreBrownianReal |
|
|
| end Problem367.ArrayInnovationLaw |
|
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| end |
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| end OAI |
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