| import OAI.Probability.FreeIsing.Martingales.ContinuousSubsequence |
| import OAI.Probability.FreeIsing.Martingales.PredictableVersion |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter Set |
| open scoped NNReal Topology |
|
|
| namespace Problem367.PredictableMesh |
|
|
| def repair {Ω : Type*} (u : ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ := |
| limUnder atTop (fun n => sample u n t ω) |
|
|
| theorem repair_isStronglyPredictable {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ} |
| (hu : StronglyAdapted F u) : IsStronglyPredictable F (repair u) := by |
| let : MeasurableSpace (ℝ≥0 × Ω) := F.predictable |
| exact StronglyMeasurable.limUnder (fun n => sample_isStronglyPredictable hu n) |
|
|
| theorem repair_eq_of_continuousOn {Ω : Type*} {u : ℝ≥0 → Ω → ℝ} |
| {T : ℝ≥0} {ω : Ω} (hc : ContinuousOn (fun t => u t ω) (Iic T)) |
| {t : ℝ≥0} (ht : t ≤ T) : repair u t ω = u t ω := by |
| apply Filter.Tendsto.limUnder_eq |
| apply (hc t ht).tendsto.comp |
| apply tendsto_nhdsWithin_iff.mpr |
| exact ⟨tendsto_left t, Filter.Eventually.of_forall |
| (fun n => (left_le n t).trans ht)⟩ |
|
|
| end Problem367.PredictableMesh |
|
|
| namespace Problem367.MartingaleSquareMaximal |
|
|
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] |
| {F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ} |
|
|
| theorem exists_predictable_continuous_subsequence_of_terminal_L2_cauchy |
| (hM : ∀ n, Martingale (M n) F μ) |
| (h2 : ∀ n t, MemLp (M n t) 2 μ) |
| (hc : ∀ n, ∀ᵐ ω ∂μ, Continuous (fun t => M n t ω)) (T : ℝ≥0) |
| (hC : CauchySeq (fun n => (h2 n T).toLp (M n T))) : |
| ∃ (v : ℕ → ℕ) (L : ℝ≥0 → Ω → ℝ), StrictMono v ∧ |
| IsStronglyPredictable F L ∧ |
| (∀ᵐ ω ∂μ, ContinuousOn (fun t => L t ω) (Iic T)) ∧ |
| (∀ᵐ ω ∂μ, ∀ t ≤ T, |
| Tendsto (fun n => M (v n) t ω) atTop (𝓝 (L t ω))) := by |
| obtain ⟨v, hv, hlim⟩ := |
| exists_continuous_subsequence_of_terminal_L2_cauchy hM h2 hc T hC |
| let R : ℝ≥0 → Ω → ℝ := fun t ω => limUnder atTop (fun n => M (v n) t ω) |
| have hR : StronglyAdapted F R := fun t => by |
| let : MeasurableSpace Ω := F t |
| exact StronglyMeasurable.limUnder (fun n => (hM (v n)).stronglyMeasurable t) |
| have hpath : ∀ᵐ ω ∂μ, ContinuousOn (fun t => R t ω) (Iic T) ∧ |
| ∀ t ≤ T, Tendsto (fun n => M (v n) t ω) atTop (𝓝 (R t ω)) := by |
| filter_upwards [hlim] with ω hω |
| obtain ⟨f, hf⟩ := hω |
| have heq (t : Iic T) : R t ω = f t := (hf t).limUnder_eq |
| constructor |
| · rw [continuousOn_iff_continuous_domRestrict] |
| exact f.continuous.congr (fun t => (heq t).symm) |
| · intro t ht |
| simpa only [heq ⟨t, ht⟩] using hf ⟨t, ht⟩ |
| refine ⟨v, PredictableMesh.repair R, hv, |
| PredictableMesh.repair_isStronglyPredictable hR, ?_, ?_⟩ |
| · filter_upwards [hpath] with ω hω |
| exact hω.1.congr (fun t ht => PredictableMesh.repair_eq_of_continuousOn hω.1 ht) |
| · filter_upwards [hpath] with ω hω |
| intro t ht |
| rw [PredictableMesh.repair_eq_of_continuousOn hω.1 ht] |
| exact hω.2 t ht |
|
|
| end Problem367.MartingaleSquareMaximal |
|
|
| end |
|
|
| end OAI |
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|
