openAI-math-dataset / lean /OAI /Probability /FreeIsing /Brownian /CoefficientApproximationIdentification.lean
| import OAI.Probability.FreeIsing.Brownian.ArrayCrossIsometry | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped NNReal Topology | |
| namespace Problem367.BrownianFiltrationGrid | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} | |
| {B : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ} | |
| (hB : IsPreBrownianReal B P) (hAd : StronglyAdapted F B) | |
| (hind : ∀ s t, s ≤ t → Indep | |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) | |
| (K : ℕ → ℝ≥0 → Ω → ℝ) (H : ℝ≥0 → Ω → ℝ) | |
| (hK : ∀ n, StronglyAdapted F (K n)) (hH : StronglyAdapted F H) | |
| (hcK : ∀ n, ∀ᵐ ω ∂P, Continuous (fun t => K n t ω)) | |
| (hcH : ∀ᵐ ω ∂P, Continuous (fun t => H t ω)) | |
| (C : ℕ → ℝ) (D : ℝ) (hC : ∀ n, 0 ≤ C n) (hD : 0 ≤ D) | |
| (hbK : ∀ n, ∀ᵐ ω ∂P, ∀ t, |K n t ω| ≤ C n) | |
| (hbH : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ D) | |
| (T : ℝ≥0) {X : ℕ → Ω → ℝ} {J : Ω → ℝ} | |
| (hX : ∀ n, MemLp (X n) 2 P) (hJ : MemLp J 2 P) | |
| (haX : ∀ n, Tendsto (fun k => ∫ ω, | |
| (BrownianGrid.uniformSum B (K n) T k ω - X n ω) ^ 2 ∂P) atTop (𝓝 0)) | |
| (haJ : Tendsto (fun k => ∫ ω, | |
| (BrownianGrid.uniformSum B H T k ω - J ω) ^ 2 ∂P) atTop (𝓝 0)) | |
| include hB hAd hind hK hH hcK hcH hC hD hbK hbH hX hJ haX haJ | |
| theorem tendsto_meanSquare_of_coefficient_interval | |
| (hcoeff : Tendsto (fun n => ∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, | |
| (K n t.toNNReal ω - H t.toNNReal ω) ^ 2 ∂P) atTop (𝓝 0)) : | |
| Tendsto (fun n => ∫ ω, (X n ω - J ω) ^ 2 ∂P) atTop (𝓝 0) := by | |
| have he (n : ℕ) := uniform_meanSquare_difference_isometry | |
| hB hAd hind (K n) H (hK n) hH (hcK n) hcH (hC n) hD | |
| (hbK n) hbH T (hX n) hJ (haX n) haJ | |
| simpa only [he] using hcoeff | |
| theorem tendsto_meanSquare_of_coefficient_timeSpace | |
| (hInt : ∀ n, Integrable (fun p : ℝ × Ω => | |
| (K n p.1.toNNReal p.2 - H p.1.toNNReal p.2) ^ 2) | |
| ((volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod P)) | |
| (hcoeff : Tendsto (fun n => ∫ p : ℝ × Ω, | |
| (K n p.1.toNNReal p.2 - H p.1.toNNReal p.2) ^ 2 | |
| ∂(volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod P) atTop (𝓝 0)) : | |
| Tendsto (fun n => ∫ ω, (X n ω - J ω) ^ 2 ∂P) atTop (𝓝 0) := by | |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure | |
| apply tendsto_meanSquare_of_coefficient_interval hB hAd hind K H hK hH hcK hcH | |
| C D hC hD hbK hbH T hX hJ haX haJ | |
| have he (n : ℕ) : | |
| (∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, | |
| (K n t.toNNReal ω - H t.toNNReal ω) ^ 2 ∂P) = | |
| ∫ p : ℝ × Ω, (K n p.1.toNNReal p.2 - H p.1.toNNReal p.2) ^ 2 | |
| ∂(volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod P := by | |
| rw [intervalIntegral.integral_of_le T.coe_nonneg] | |
| exact (integral_prod _ (hInt n)).symm | |
| simpa only [he] using hcoeff | |
| theorem ae_eq_of_coefficient_timeSpace_approximation | |
| (hInt : ∀ n, Integrable (fun p : ℝ × Ω => | |
| (K n p.1.toNNReal p.2 - H p.1.toNNReal p.2) ^ 2) | |
| ((volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod P)) | |
| (hcoeff : Tendsto (fun n => ∫ p : ℝ × Ω, | |
| (K n p.1.toNNReal p.2 - H p.1.toNNReal p.2) ^ 2 | |
| ∂(volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod P) atTop (𝓝 0)) | |
| {I : Ω → ℝ} (hI : MemLp I 2 P) | |
| (hlimI : Tendsto (fun n => ∫ ω, (X n ω - I ω) ^ 2 ∂P) atTop (𝓝 0)) : | |
| I =ᵐ[P] J := by | |
| exact BrownianGrid.ae_eq_of_mean_square_limits hX hI hJ hlimI | |
| (tendsto_meanSquare_of_coefficient_timeSpace hB hAd hind K H hK hH hcK hcH | |
| C D hC hD hbK hbH T hX hJ haX haJ hInt hcoeff) | |
| end Problem367.BrownianFiltrationGrid | |
| end | |
| end OAI | |