| import OAI.Probability.FreeIsing.Factors.ContinuousRecovery |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter TopologicalSpace |
| open scoped NNReal Topology |
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| namespace Problem367.StoppedProcessTransport |
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| variable {Ω Ω' ι E : Type*} |
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| theorem stoppedProcess_pullback [Nonempty ι] [LinearOrder ι] |
| (φ : Ω' → Ω) (X : ι → Ω → E) (τ : Ω → WithTop ι) : |
| stoppedProcess (fun t => X t ∘ φ) (τ ∘ φ) = |
| fun t => stoppedProcess X τ t ∘ φ := rfl |
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| theorem stoppedValue_pullback [Nonempty ι] |
| (φ : Ω' → Ω) (X : ι → Ω → E) (τ : Ω → WithTop ι) : |
| stoppedValue (fun t => X t ∘ φ) (τ ∘ φ) = stoppedValue X τ ∘ φ := rfl |
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| theorem isStoppingTime_pullback [Preorder ι] |
| [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] |
| {F : Filtration ι mΩ} {G : Filtration ι mΩ'} |
| {φ : Ω' → Ω} (hφ : ∀ t, Measurable[G t, F t] φ) |
| {τ : Ω → WithTop ι} (hτ : IsStoppingTime F τ) : |
| IsStoppingTime G (τ ∘ φ) := by |
| intro t |
| exact (hφ t) (hτ t) |
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| theorem stoppedValue_ae_eq_of_continuous |
| [MeasurableSpace Ω] {P : Measure Ω} |
| [TopologicalSpace ι] [SeparableSpace ι] [Nonempty ι] |
| [TopologicalSpace E] [T2Space E] |
| {X Y : ι → Ω → E} |
| (hX : ∀ᵐ ω ∂P, Continuous (fun t => X t ω)) |
| (hY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω)) |
| (hXY : ∀ t, X t =ᵐ[P] Y t) (τ : Ω → WithTop ι) : |
| stoppedValue X τ =ᵐ[P] stoppedValue Y τ := by |
| filter_upwards [ae_eq_of_continuous_of_evaluation_ae_eq hX hY hXY] with ω hω |
| exact congrFun hω (τ ω).untopA |
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| theorem stoppedProcess_ae_eq_of_continuous |
| [MeasurableSpace Ω] {P : Measure Ω} |
| [TopologicalSpace ι] [SeparableSpace ι] [Nonempty ι] [LinearOrder ι] |
| [TopologicalSpace E] [T2Space E] |
| {X Y : ι → Ω → E} |
| (hX : ∀ᵐ ω ∂P, Continuous (fun t => X t ω)) |
| (hY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω)) |
| (hXY : ∀ t, X t =ᵐ[P] Y t) (τ : Ω → WithTop ι) : |
| (fun ω t => stoppedProcess X τ t ω) =ᵐ[P] |
| (fun ω t => stoppedProcess Y τ t ω) := by |
| filter_upwards [ae_eq_of_continuous_of_evaluation_ae_eq hX hY hXY] with ω hω |
| funext t |
| exact congrFun hω (min (t : WithTop ι) (τ ω)).untopA |
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| theorem integral_stoppedProcess_sq_pullback |
| [MeasurableSpace Ω] [MeasurableSpace Ω'] |
| [Nonempty ι] [LinearOrder ι] |
| {P : Measure Ω} {Q : Measure Ω'} {φ : Ω' → Ω} |
| (hφ : MeasurePreserving φ Q P) (X : ι → Ω → ℝ) |
| (τ : Ω → WithTop ι) (t : ι) |
| (hX : AEStronglyMeasurable (stoppedProcess X τ t) P) : |
| (∫ ω, stoppedProcess (fun s => X s ∘ φ) (τ ∘ φ) t ω ^ 2 ∂Q) = |
| ∫ ω, stoppedProcess X τ t ω ^ 2 ∂P := by |
| change (∫ ω, stoppedProcess X τ t (φ ω) ^ 2 ∂Q) = _ |
| have hm : AEStronglyMeasurable (fun ω => stoppedProcess X τ t ω ^ 2) (Q.map φ) := by |
| rw [hφ.map_eq] |
| exact hX.pow 2 |
| simpa only [hφ.map_eq] using (integral_map hφ.aemeasurable hm).symm |
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| theorem integral_stoppedProcess_sq_congr |
| [MeasurableSpace Ω] {P : Measure Ω} |
| [TopologicalSpace ι] [SeparableSpace ι] [Nonempty ι] [LinearOrder ι] |
| {X Y : ι → Ω → ℝ} |
| (hX : ∀ᵐ ω ∂P, Continuous (fun t => X t ω)) |
| (hY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω)) |
| (hXY : ∀ t, X t =ᵐ[P] Y t) (τ : Ω → WithTop ι) (t : ι) : |
| (∫ ω, stoppedProcess X τ t ω ^ 2 ∂P) = |
| ∫ ω, stoppedProcess Y τ t ω ^ 2 ∂P := by |
| apply integral_congr_ae |
| filter_upwards [stoppedProcess_ae_eq_of_continuous hX hY hXY τ] with ω hω |
| exact congrArg (fun x : ℝ => x ^ 2) (congrFun hω t) |
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| theorem integral_before_stop_pullback |
| [mΩ : MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} {Q : Measure Ω'} {φ : Ω' → Ω} |
| (hφ : MeasurePreserving φ Q P) |
| {F : Filtration ℝ≥0 mΩ} {τ : Ω → WithTop ℝ≥0} |
| (hτ : IsStoppingTime F τ) {A : Ω → ℝ} |
| (hA : AEStronglyMeasurable A P) (s : ℝ≥0) : |
| (∫ ω, if (s : WithTop ℝ≥0) ≤ τ (φ ω) then A (φ ω) else 0 ∂Q) = |
| ∫ ω, if (s : WithTop ℝ≥0) ≤ τ ω then A ω else 0 ∂P := by |
| classical |
| have hs : MeasurableSet {ω | (s : WithTop ℝ≥0) ≤ τ ω} := |
| F.le s _ (hτ.measurableSet_ge s) |
| have hm : AEStronglyMeasurable |
| (fun ω => if (s : WithTop ℝ≥0) ≤ τ ω then A ω else 0) (Q.map φ) := by |
| rw [hφ.map_eq] |
| apply (hA.indicator hs).congr |
| exact Filter.Eventually.of_forall |
| (fun ω => Set.indicator_apply {ω | (s : WithTop ℝ≥0) ≤ τ ω} A ω) |
| simpa only [hφ.map_eq] using (integral_map hφ.aemeasurable hm).symm |
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| theorem intervalIntegral_before_stop_pullback |
| [mΩ : MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} {Q : Measure Ω'} {φ : Ω' → Ω} |
| (hφ : MeasurePreserving φ Q P) |
| {F : Filtration ℝ≥0 mΩ} {τ : Ω → WithTop ℝ≥0} |
| (hτ : IsStoppingTime F τ) {A : ℝ → Ω → ℝ} |
| (hA : ∀ s, AEStronglyMeasurable (A s) P) (a b : ℝ) : |
| (∫ s in a..b, ∫ ω, |
| if (s.toNNReal : WithTop ℝ≥0) ≤ τ (φ ω) then A s (φ ω) else 0 ∂Q) = |
| ∫ s in a..b, ∫ ω, |
| if (s.toNNReal : WithTop ℝ≥0) ≤ τ ω then A s ω else 0 ∂P := by |
| apply intervalIntegral.integral_congr |
| intro s _ |
| exact integral_before_stop_pullback hφ hτ (hA s) s.toNNReal |
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| theorem local_square_identity_congr |
| [MeasurableSpace Ω] {P : Measure Ω} |
| {X Y : ℝ≥0 → Ω → ℝ} |
| (hX : ∀ᵐ ω ∂P, Continuous (fun t => X t ω)) |
| (hY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω)) |
| (hXY : ∀ t, X t =ᵐ[P] Y t) |
| {D E : ℝ → Ω → ℝ} (hDE : ∀ s, D s =ᵐ[P] E s) |
| (τ : Ω → WithTop ℝ≥0) (t : ℝ≥0) |
| (hlocal : (∫ ω, stoppedProcess X τ t ω ^ 2 ∂P) = |
| (∫ ω, X 0 ω ^ 2 ∂P) + ∫ s in (0 : ℝ)..(t : ℝ), |
| ∫ ω, if (s.toNNReal : WithTop ℝ≥0) ≤ τ ω then D s ω else 0 ∂P) : |
| (∫ ω, stoppedProcess Y τ t ω ^ 2 ∂P) = |
| (∫ ω, Y 0 ω ^ 2 ∂P) + ∫ s in (0 : ℝ)..(t : ℝ), |
| ∫ ω, if (s.toNNReal : WithTop ℝ≥0) ≤ τ ω then E s ω else 0 ∂P := by |
| rw [← integral_stoppedProcess_sq_congr hX hY hXY τ t, hlocal] |
| congr 1 |
| · apply integral_congr_ae |
| exact (hXY 0).mono (fun ω hω => congrArg (fun x : ℝ => x ^ 2) hω) |
| · apply intervalIntegral.integral_congr |
| intro s _ |
| apply integral_congr_ae |
| filter_upwards [hDE s] with ω hω |
| simp only [hω] |
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| end Problem367.StoppedProcessTransport |
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| end |
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| end OAI |
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