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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | 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
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