| import OAI.Probability.FreeIsing.Brownian.ArrayMoments |
| import OAI.Probability.FreeIsing.Brownian.Innovation |
| import OAI.Probability.FreeIsing.Estimates.ProgressiveArrayInnovation |
|
|
| namespace OAI |
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| noncomputable section |
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|
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
|
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| namespace Problem367 |
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| theorem ae_array_innovation_increment_drift_bound |
| {ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω] |
| (ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν] |
| [∀ j, IsProbabilityMeasure (μ j)] |
| {B : ι → ℝ≥0 → Ω → ℝ} (hmB : ∀ i t, Measurable (B i t)) |
| {σ : S → ι → ℝ} (hσ : Measurable σ) (i : ι) |
| {C : ℝ≥0} (hσbound : ∀ᵐ s ∂ν, |σ s i| ≤ (C : ℝ)) |
| {m : ℝ≥0 → S × (ι → Ω) → ℝ} |
| (hm : IsStronglyProgressive (arrayObservationFiltration B hmB σ hσ) m) |
| (hposterior : ∀ t, m 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]) |
| {s t : ℝ≥0} (hst : s ≤ t) : |
| ∀ᵐ q ∂ν.prod (Measure.infinitePi μ), |
| |((plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q) - |
| (plantedArrayObservation B σ i s q - nnrealTimePrimitive m s q)) - |
| (B i t (q.2 i) - B i s (q.2 i))| ≤ |
| (2 * (C : ℝ)) * ((t : ℝ) - (s : ℝ)) := by |
| have hsignal : ∀ᵐ q ∂ν.prod (Measure.infinitePi μ), |σ q.1 i| ≤ (C : ℝ) := |
| Measure.quasiMeasurePreserving_fst.ae hσbound |
| filter_upwards [hsignal, |
| ae_lipschitzWith_array_posterior_primitive ν μ hmB hσ i hσbound hm hposterior] |
| with q hsq hlip |
| have htime : 0 ≤ (t : ℝ) - (s : ℝ) := sub_nonneg.mpr hst |
| have hprim : |nnrealTimePrimitive m t q - nnrealTimePrimitive m s q| ≤ |
| (C : ℝ) * ((t : ℝ) - (s : ℝ)) := by |
| simpa only [Real.dist_eq, NNReal.dist_eq, abs_of_nonneg htime] using |
| hlip.dist_le_mul t s |
| have heq : ((plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q) - |
| (plantedArrayObservation B σ i s q - nnrealTimePrimitive m s q)) - |
| (B i t (q.2 i) - B i s (q.2 i)) = |
| ((t : ℝ) - (s : ℝ)) * σ q.1 i - |
| (nnrealTimePrimitive m t q - nnrealTimePrimitive m s q) := by |
| unfold plantedArrayObservation |
| ring |
| rw [heq] |
| calc |
| _ ≤ |((t : ℝ) - (s : ℝ)) * σ q.1 i| + |
| |nnrealTimePrimitive m t q - nnrealTimePrimitive m s q| := abs_sub _ _ |
| _ ≤ ((t : ℝ) - (s : ℝ)) * (C : ℝ) + |
| (C : ℝ) * ((t : ℝ) - (s : ℝ)) := by |
| rw [abs_mul, abs_of_nonneg htime] |
| exact add_le_add (mul_le_mul_of_nonneg_left hsq htime) hprim |
| _ = _ := by ring |
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|
| theorem preBrownian_array_innovation_of_progressive_posterior |
| {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) (i : ι) |
| {C : ℝ≥0} (hσbound : ∀ᵐ s ∂ν, |σ s i| ≤ (C : ℝ)) |
| {m : ℝ≥0 → S × (ι → Ω) → ℝ} |
| (hm : IsStronglyProgressive (arrayObservationFiltration B hmB σ hσ) m) |
| (hposterior : ∀ t, m 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]) : |
| IsPreBrownianReal |
| (fun t q => plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q) |
| (ν.prod (Measure.infinitePi μ)) := by |
| classical |
| have (j : ι) : IsProbabilityMeasure (μ j) := |
| (hB j).isGaussianProcess.isProbabilityMeasure |
| apply BrownianInnovation.isPreBrownianReal_of_bounded_drift |
| (planted_coordinate_preBrownian ν μ hB i) |
| (preBrownian_array_innovation_martingale_of_progressive_posterior |
| ν μ hB hmB hσ i hσbound hm hposterior) |
| (L := 2 * (C : ℝ)) |
| · intro s t hst |
| have hv := planted_weighted_increment_sq_condExp_subfield ν μ hB hmB hst |
| ({i} : Finset ι) (fun _ => (1 : ℝ)) |
| (arrayObservationFiltration_le_signalArrayFiltration B hmB σ hσ s) |
| simpa using hv |
| · positivity |
| · intro s t hst |
| exact ae_array_innovation_increment_drift_bound ν μ hmB hσ i hσbound hm hposterior hst |
| · have hzero := (planted_noise_coordinate_measurePreserving ν μ i).quasiMeasurePreserving.ae |
| (hB i).eval_zero_ae_eq_zero |
| filter_upwards [hzero] with q hq |
| simpa only [plantedArrayObservation, nnrealTimePrimitive, NNReal.coe_zero, |
| zero_mul, add_zero, intervalIntegral.integral_same, sub_zero] using hq |
|
|
| theorem brownian_array_innovation_of_progressive_posterior |
| {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, IsBrownianReal (B i) (μ i)) |
| (hmB : ∀ i t, Measurable (B i t)) |
| {σ : S → ι → ℝ} (hσ : Measurable σ) (i : ι) |
| {C : ℝ≥0} (hσbound : ∀ᵐ s ∂ν, |σ s i| ≤ (C : ℝ)) |
| {m : ℝ≥0 → S × (ι → Ω) → ℝ} |
| (hm : IsStronglyProgressive (arrayObservationFiltration B hmB σ hσ) m) |
| (hposterior : ∀ t, m 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]) : |
| IsBrownianReal |
| (fun t q => plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q) |
| (ν.prod (Measure.infinitePi μ)) := |
| ⟨preBrownian_array_innovation_of_progressive_posterior |
| ν μ (fun j => (hB j).toIsPreBrownianReal) hmB hσ i hσbound hm hposterior, |
| (brownian_array_innovation_regular_martingale ν μ hB hmB hσ i hσbound hm hposterior).2.1⟩ |
|
|
| end Problem367 |
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| end |
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| end OAI |
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