| import OAI.Probability.FreeIsing.Brownian.Martingale |
| import OAI.Probability.FreeIsing.Estimates.IndependentEnlargement |
| import OAI.Probability.FreeIsing.Martingales.InnovationMartingale |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
|
|
| namespace Problem367 |
|
|
| def brownianSignalFiltration {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, StronglyMeasurable (B t)) |
| (σ : Ω → ℝ) (hσ : Measurable σ) : Filtration ℝ≥0 mΩ := |
| Filtration.natural B hm ⊔ Filtration.const ℝ≥0 (MeasurableSpace.comap σ inferInstance) hσ.comap_le |
|
|
| theorem preBrownian_martingale_signalFiltration |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hm : ∀ t, StronglyMeasurable (B t)) {σ : Ω → ℝ} (hσ : Measurable σ) |
| (hind : IndepFun (fun ω t => B t ω) σ P) : |
| Martingale B (brownianSignalFiltration B hm σ hσ) P := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hfull : Measurable (fun ω t => B t ω) := Measurable.of_eval (fun t => (hm t).measurable) |
| have hcoord (t : ℝ≥0) : |
| @Measurable Ω ℝ (MeasurableSpace.comap (fun ω t => B t ω) inferInstance) inferInstance (B t) := |
| by |
| have harray : @Measurable Ω (ℝ≥0 → ℝ) |
| (MeasurableSpace.comap (fun ω t => B t ω) inferInstance) inferInstance |
| (fun ω t => B t ω) := measurable_iff_comap_le.mpr le_rfl |
| exact (show Measurable (fun f : ℝ≥0 → ℝ => f t) from measurable_pi_apply t).comp harray |
| have hNat (t : ℝ≥0) : Filtration.natural B hm t ≤ |
| MeasurableSpace.comap (fun ω t => B t ω) inferInstance := by |
| change (⨆ s ≤ t, MeasurableSpace.comap (B s) inferInstance) ≤ _ |
| exact iSup₂_le fun s _ => (hcoord s).comap_le |
| exact martingale_independent_enlargement hfull.comap_le hσ.comap_le hNat hind |
| (preBrownian_martingale_natural hB hm) |
|
|
| def plantedObservation {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (σ : Ω → ℝ) |
| (t : ℝ≥0) (ω : Ω) : ℝ := B t ω + (t : ℝ) * σ ω |
|
|
| theorem plantedObservation_stronglyMeasurable |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} |
| (hm : ∀ t, StronglyMeasurable (B t)) {σ : Ω → ℝ} (hσ : Measurable σ) (t : ℝ≥0) : |
| StronglyMeasurable (plantedObservation B σ t) := |
| (hm t).add (stronglyMeasurable_const.mul hσ.stronglyMeasurable) |
|
|
| def observationFiltration {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, StronglyMeasurable (B t)) |
| (σ : Ω → ℝ) (hσ : Measurable σ) : Filtration ℝ≥0 mΩ := |
| Filtration.natural (plantedObservation B σ) (plantedObservation_stronglyMeasurable hm hσ) |
|
|
| theorem naturalFiltration_le_of_stronglyAdapted |
| {Ω ι : Type*} [Preorder ι] [mΩ : MeasurableSpace Ω] |
| {X : ι → Ω → ℝ} (hm : ∀ t, StronglyMeasurable (X t)) |
| {G : Filtration ι mΩ} (had : StronglyAdapted G X) : |
| Filtration.natural X hm ≤ G := by |
| intro t |
| change (⨆ s ≤ t, MeasurableSpace.comap (X s) inferInstance) ≤ G t |
| exact iSup₂_le fun s hst => ((had s).mono (G.mono hst)).measurable.comap_le |
|
|
| theorem observationFiltration_le_signalFiltration |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} |
| (hm : ∀ t, StronglyMeasurable (B t)) {σ : Ω → ℝ} (hσ : Measurable σ) : |
| observationFiltration B hm σ hσ ≤ brownianSignalFiltration B hm σ hσ := by |
| apply naturalFiltration_le_of_stronglyAdapted |
| intro t |
| have hBt : StronglyMeasurable[brownianSignalFiltration B hm σ hσ t] (B t) := |
| (Filtration.stronglyAdapted_natural hm t).mono le_sup_left |
| have hσm : StronglyMeasurable[MeasurableSpace.comap σ inferInstance] σ := |
| (measurable_iff_comap_le.mpr le_rfl).stronglyMeasurable |
| have hσG : StronglyMeasurable[brownianSignalFiltration B hm σ hσ t] σ := |
| hσm.mono le_sup_right |
| exact hBt.add (stronglyMeasurable_const.mul hσG) |
|
|
| theorem preBrownian_observation_innovation_martingale |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B m : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hmB : ∀ t, StronglyMeasurable (B t)) {σ : Ω → ℝ} |
| (hσ : Measurable σ) (hσint : Integrable σ P) |
| (hind : IndepFun (fun ω t => B t ω) σ P) |
| (hmAd : StronglyAdapted (observationFiltration B hmB σ hσ) m) |
| (hposterior : ∀ t, m t =ᵐ[P] P[σ | observationFiltration B hmB σ hσ t]) |
| (hAad : StronglyAdapted (observationFiltration B hmB σ hσ) (nnrealTimePrimitive m)) |
| (hInt : ∀ s t : ℝ≥0, s ≤ t → |
| Integrable (fun p : ℝ × Ω => m p.1.toNNReal p.2) |
| ((volume.restrict (Set.Ioc (s : ℝ) (t : ℝ))).prod P)) : |
| Martingale (fun t ω => plantedObservation B σ t ω - nnrealTimePrimitive m t ω) |
| (observationFiltration B hmB σ hσ) P := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| exact compensated_observation_martingale_of_posterior_version |
| (preBrownian_martingale_signalFiltration hB hmB hσ hind) |
| (observationFiltration_le_signalFiltration hmB hσ) hσint |
| (Filtration.stronglyAdapted_natural (plantedObservation_stronglyMeasurable hmB hσ)) |
| hmAd hposterior hAad hInt |
|
|
| end Problem367 |
|
|
| end OAI |
|
|
