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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | 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
end
end OAI
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