| import OAI.Probability.FreeIsing.Brownian.FiltrationIntegral |
| import OAI.Probability.FreeIsing.Brownian.UniformIsometry |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
|
|
| namespace Problem367.BrownianFiltrationGrid |
| open BrownianGrid |
|
|
| theorem preBrownian_difference_isometry |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) |
| (t : ℕ → ℝ≥0) (ht : Monotone t) (H K : ℕ → Ω → ℝ) |
| {k : ℕ} (hk : 0 < k) |
| (hH : ∀ j, StronglyMeasurable[F (t (j * k))] (H j)) |
| (hK : ∀ i, StronglyMeasurable[F (t i)] (K i)) |
| (hH2 : ∀ j, MemLp (H j) 2 P) (hK2 : ∀ i, MemLp (K i) 2 P) (n : ℕ) : |
| (∫ ω, (integralSum B (fun j => t (j * k)) H n ω - |
| integralSum B t K (n * k) ω) ^ 2 ∂P) = |
| ∑ i ∈ Finset.range (n * k), ((t (i + 1) : ℝ) - (t i : ℝ)) * |
| ∫ ω, (H (i / k) ω - K i ω) ^ 2 ∂P := by |
| rw [integralSum_refinement B t H n hk] |
| simpa only [integralSum, sub_mul, Finset.sum_sub_distrib] using |
| BrownianFiltration.predictable_sum_isometry hB hAd hind t ht |
| (fun i ω => H (i / k) ω - K i ω) |
| (fun i => ((hH (i / k)).mono (F.mono (ht (Nat.div_mul_le_self i k)))).sub (hK i)) |
| (fun i => (hH2 (i / k)).sub (hK2 i)) (Finset.range (n * k)) |
|
|
| theorem preBrownian_uniform_difference_isometry |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) |
| (T : ℝ≥0) {n k : ℕ} (hn : 0 < n) (hk : 0 < k) : |
| (∫ ω, (uniformSum B H T n ω - uniformSum B H T (n * k) ω) ^ 2 ∂P) = |
| ∑ i ∈ Finset.range (n * k), |
| ((uniformGrid T (n * k) (i + 1) : ℝ) - (uniformGrid T (n * k) i : ℝ)) * |
| ∫ ω, (H (uniformGrid T n (i / k)) ω - H (uniformGrid T (n * k) i) ω) ^ 2 ∂P := by |
| have hgrid : (fun j => uniformGrid T (n * k) (j * k)) = uniformGrid T n := by |
| funext j |
| exact uniformGrid_refine T hn hk j |
| have hcoeff : ∀ j, StronglyMeasurable[ |
| F (uniformGrid T (n * k) (j * k))] |
| (H (uniformGrid T n j)) := by |
| intro j |
| rw [uniformGrid_refine T hn hk] |
| exact hH _ |
| simpa only [hgrid, uniformSum] using preBrownian_difference_isometry hB hAd hind |
| (uniformGrid T (n * k)) (uniformGrid_mono T (n * k)) |
| (fun j => H (uniformGrid T n j)) (fun i => H (uniformGrid T (n * k) i)) hk |
| hcoeff (fun i => hH _) (fun j => hH2 _) (fun i => hH2 _) n |
|
|
| theorem preBrownian_uniform_difference_bound |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) |
| (T : ℝ≥0) {n k : ℕ} (hn : 0 < n) (hk : 0 < k) {ε : ℝ} |
| (happrox : ∀ i < n * k, |
| (∫ ω, (H (uniformGrid T n (i / k)) ω - |
| H (uniformGrid T (n * k) i) ω) ^ 2 ∂P) ≤ ε) : |
| (∫ ω, (uniformSum B H T n ω - uniformSum B H T (n * k) ω) ^ 2 ∂P) ≤ |
| (T : ℝ) * ε := by |
| rw [preBrownian_uniform_difference_isometry hB hAd hind H hH hH2 T hn hk] |
| calc |
| _ ≤ ∑ i ∈ Finset.range (n * k), |
| ((uniformGrid T (n * k) (i + 1) : ℝ) - (uniformGrid T (n * k) i : ℝ)) * ε := by |
| apply Finset.sum_le_sum |
| intro i hi |
| apply mul_le_mul_of_nonneg_left (happrox i (Finset.mem_range.mp hi)) |
| exact sub_nonneg.mpr (NNReal.coe_le_coe.mpr (uniformGrid_mono T (n * k) (Nat.le_succ i))) |
| _ = (T : ℝ) * ε := by |
| rw [← Finset.sum_mul, Finset.sum_range_sub (fun i => (uniformGrid T (n * k) i : ℝ))] |
| simp [uniformGrid_end T (Nat.mul_pos hn hk)] |
|
|
| theorem integralSum_memLp |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (_hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) |
| (t : ℕ → ℝ≥0) (ht : Monotone t) (H : ℕ → Ω → ℝ) |
| (hH : ∀ i, StronglyMeasurable[F (t i)] (H i)) |
| (hH2 : ∀ i, MemLp (H i) 2 P) (n : ℕ) : |
| MemLp (integralSum B t H n) 2 P := by |
| apply memLp_finsetSum |
| intro i hi |
| exact BrownianFiltration.predictable_mul_memLp_two hB (hind _ _ (ht (Nat.le_succ i))) (hH i) (hH2 i) |
|
|
| theorem uniformSum_memLp |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) (T : ℝ≥0) (n : ℕ) : |
| MemLp (uniformSum B H T n) 2 P := |
| integralSum_memLp hB hAd hind (uniformGrid T n) (uniformGrid_mono T n) |
| (fun i => H (uniformGrid T n i)) (fun _ => hH _) (fun _ => hH2 _) n |
|
|
| theorem preBrownian_uniform_difference_bound_of_modulus |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) |
| (T : ℝ≥0) {n k : ℕ} (hn : 0 < n) (hk : 0 < k) {ε δ : ℝ} |
| (hmesh : (T : ℝ) / (n : ℝ) ≤ δ) |
| (hmod : ∀ s t : ℝ≥0, s ≤ t → t ≤ T → (t : ℝ) - (s : ℝ) ≤ δ → |
| (∫ ω, (H s ω - H t ω) ^ 2 ∂P) ≤ ε) : |
| (∫ ω, (uniformSum B H T n ω - uniformSum B H T (n * k) ω) ^ 2 ∂P) ≤ |
| (T : ℝ) * ε := by |
| apply preBrownian_uniform_difference_bound hB hAd hind H hH hH2 T hn hk |
| intro i hi |
| apply hmod |
| · exact uniformGrid_floor_le T hn hk i |
| · exact uniformGrid_le_horizon T (Nat.mul_pos hn hk) hi.le |
| · exact (uniformGrid_floor_sub_le T hn hk i).trans hmesh |
|
|
| theorem preBrownian_uniform_pair_bound |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) |
| (T : ℝ≥0) {n m : ℕ} (hn : 0 < n) (hmn : 0 < m) {ε δ : ℝ} |
| (hmeshN : (T : ℝ) / (n : ℝ) ≤ δ) (hmeshM : (T : ℝ) / (m : ℝ) ≤ δ) |
| (hmod : ∀ s t : ℝ≥0, s ≤ t → t ≤ T → (t : ℝ) - (s : ℝ) ≤ δ → |
| (∫ ω, (H s ω - H t ω) ^ 2 ∂P) ≤ ε) : |
| (∫ ω, (uniformSum B H T n ω - uniformSum B H T m ω) ^ 2 ∂P) ≤ |
| 4 * (T : ℝ) * ε := by |
| have hN := preBrownian_uniform_difference_bound_of_modulus hB hAd hind H hH hH2 T hn hmn hmeshN hmod |
| have hM := preBrownian_uniform_difference_bound_of_modulus hB hAd hind H hH hH2 T hmn hn hmeshM hmod |
| rw [Nat.mul_comm m n] at hM |
| have htriangle := integral_sq_sub_le_two (uniformSum_memLp hB hAd hind H hH hH2 T n) |
| (uniformSum_memLp hB hAd hind H hH hH2 T m) (uniformSum_memLp hB hAd hind H hH hH2 T (n * m)) |
| linarith |
|
|
| def uniformSumLp |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) (T : ℝ≥0) (n : ℕ) : Lp ℝ 2 P := |
| (uniformSum_memLp hB hAd hind H hH hH2 T n).toLp (uniformSum B H T n) |
|
|
| lemma coe_uniformSumLp |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) (T : ℝ≥0) (n : ℕ) : |
| (uniformSumLp hB hAd hind H hH hH2 T n : Ω → ℝ) =ᵐ[P] uniformSum B H T n := |
| (uniformSum_memLp hB hAd hind H hH hH2 T n).coeFn_toLp |
|
|
| theorem cauchySeq_uniformSumLp |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) (T : ℝ≥0) |
| (hcont : ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), |
| ∀ s t : ℝ≥0, s ≤ t → t ≤ T → (t : ℝ) - (s : ℝ) ≤ δ → |
| (∫ ω, (H s ω - H t ω) ^ 2 ∂P) ≤ ε) : |
| CauchySeq (uniformSumLp hB hAd hind H hH hH2 T) := by |
| apply Metric.cauchySeq_iff.mpr |
| intro ε hε |
| let η := ε ^ 2 / (4 * (T : ℝ) + 1) |
| have hden : 0 < 4 * (T : ℝ) + 1 := by positivity |
| have hη : 0 < η := div_pos (sq_pos_of_pos hε) hden |
| obtain ⟨δ, hδ, hmod⟩ := hcont η hη |
| obtain ⟨N, hN⟩ := exists_nat_gt ((T : ℝ) / δ) |
| have hmesh (n : ℕ) (hn : N + 1 ≤ n) : (T : ℝ) / (n : ℝ) ≤ δ := by |
| have hn0 : 0 < n := by omega |
| have hnR : (0 : ℝ) < n := by exact_mod_cast hn0 |
| apply (div_le_iff₀ hnR).2 |
| have hN' : (T : ℝ) < (N : ℝ) * δ := (div_lt_iff₀ hδ).mp hN |
| have hNn : (N : ℝ) ≤ (n : ℝ) := by exact_mod_cast (show N ≤ n by omega) |
| nlinarith |
| refine ⟨N + 1, ?_⟩ |
| intro n hn m hmn |
| have hn0 : 0 < n := by omega |
| have hm0 : 0 < m := by omega |
| have hb := preBrownian_uniform_pair_bound hB hAd hind H hH hH2 T hn0 hm0 |
| (hmesh n hn) (hmesh m hmn) hmod |
| have hnorm : ‖uniformSumLp hB hAd hind H hH hH2 T n - uniformSumLp hB hAd hind H hH hH2 T m‖ ^ 2 ≤ |
| 4 * (T : ℝ) * η := by |
| rw [uniformSumLp, uniformSumLp, norm_sq_toLp_sub] |
| exact hb |
| have hsmall : 4 * (T : ℝ) * η < ε ^ 2 := by |
| dsimp [η] |
| rw [← mul_div_assoc] |
| apply (div_lt_iff₀ hden).2 |
| nlinarith [sq_pos_of_pos hε] |
| rw [dist_eq_norm] |
| nlinarith [norm_nonneg |
| (uniformSumLp hB hAd hind H hH hH2 T n - uniformSumLp hB hAd hind H hH hH2 T m)] |
| |
| theorem exists_uniformSumLp_limit |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) (T : ℝ≥0) |
| (hcont : ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), |
| ∀ s t : ℝ≥0, s ≤ t → t ≤ T → (t : ℝ) - (s : ℝ) ≤ δ → |
| (∫ ω, (H s ω - H t ω) ^ 2 ∂P) ≤ ε) : |
| ∃ I : Lp ℝ 2 P, Tendsto (uniformSumLp hB hAd hind H hH hH2 T) atTop (𝓝 I) := |
| cauchySeq_tendsto_of_complete (cauchySeq_uniformSumLp hB hAd hind H hH hH2 T hcont) |
| |
| theorem exists_uniform_integral |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) (T : ℝ≥0) |
| (hcont : ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), |
| ∀ s t : ℝ≥0, s ≤ t → t ≤ T → (t : ℝ) - (s : ℝ) ≤ δ → |
| (∫ ω, (H s ω - H t ω) ^ 2 ∂P) ≤ ε) : |
| ∃ I : Ω → ℝ, StronglyMeasurable I ∧ MemLp I 2 P ∧ |
| Tendsto (fun n => ∫ ω, (uniformSum B H T n ω - I ω) ^ 2 ∂P) atTop (𝓝 0) := by |
| obtain ⟨I, hI⟩ := exists_uniformSumLp_limit hB hAd hind H hH hH2 T hcont |
| refine ⟨I, Lp.stronglyMeasurable I, Lp.memLp I, ?_⟩ |
| have hnorm := (hI.sub (tendsto_const_nhds (x := I))).norm.pow 2 |
| simp only [sub_self, norm_zero, zero_pow (by norm_num : (2 : ℕ) ≠ 0)] at hnorm |
| apply hnorm.congr' |
| apply Filter.Eventually.of_forall |
| intro n |
| dsimp only |
| rw [norm_sq_L2_eq_integral] |
| apply integral_congr_ae |
| filter_upwards [Lp.coeFn_sub (uniformSumLp hB hAd hind H hH hH2 T n) I, |
| coe_uniformSumLp hB hAd hind H hH hH2 T n] with ω hsub hsum |
| simp only [Pi.sub_apply] at hsub |
| rw [hsub, hsum] |
|
|
| theorem exists_uniform_integral_of_bounded_continuous |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hc : ∀ᵐ ω ∂P, Continuous (fun t => H t ω)) |
| {C : ℝ} (hC : 0 ≤ C) (hb : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C) |
| (T : ℝ≥0) : |
| ∃ I : Ω → ℝ, StronglyMeasurable I ∧ MemLp I 2 P ∧ |
| Tendsto (fun n => ∫ ω, (uniformSum B H T n ω - I ω) ^ 2 ∂P) atTop (𝓝 0) := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hHm : ∀ t, AEStronglyMeasurable (H t) P := fun t => |
| ((hH t).mono (F.le t)).aestronglyMeasurable |
| have hH2 : ∀ t, MemLp (H t) 2 P := by |
| intro t |
| apply MemLp.of_bound (hHm t) C |
| filter_upwards [hb] with ω hω |
| simpa only [Real.norm_eq_abs] using hω t |
| apply exists_uniform_integral hB hAd hind H hH hH2 T |
| intro ε hε |
| exact MeanSquareContinuity.uniform_integral_sq_sub_le hHm hc hC hb T hε |
|
|
| theorem uniformSumLp_norm_sq |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) (T : ℝ≥0) (n : ℕ) : |
| ‖uniformSumLp hB hAd hind H hH hH2 T n‖ ^ 2 = |
| ((T : ℝ) / n) * ∑ i ∈ Finset.range n, ∫ ω, (H (uniformGrid T n i) ω) ^ 2 ∂P := by |
| calc |
| _ = ∫ ω, (uniformSum B H T n ω) ^ 2 ∂P := by |
| rw [norm_sq_L2_eq_integral] |
| apply integral_congr_ae |
| filter_upwards [coe_uniformSumLp hB hAd hind H hH hH2 T n] with ω hω |
| rw [hω] |
| _ = ∑ i ∈ Finset.range n, |
| ((uniformGrid T n (i + 1) : ℝ) - (uniformGrid T n i : ℝ)) * |
| ∫ ω, (H (uniformGrid T n i) ω) ^ 2 ∂P := |
| BrownianFiltration.predictable_sum_isometry hB hAd hind (uniformGrid T n) (uniformGrid_mono T n) |
| (fun i => H (uniformGrid T n i)) (fun i => hH _) (fun i => hH2 _) (Finset.range n) |
| _ = _ := by simp_rw [uniformGrid_step]; rw [← Finset.mul_sum] |
|
|
| theorem uniformSumLp_limit_isometry |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| {F : Filtration ℝ≥0 mΩ} (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (F) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) |
| (hc : ∀ᵐ ω ∂P, Continuous (fun t => H t ω)) |
| {C : ℝ} (hC : 0 ≤ C) (hb : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C) |
| (T : ℝ≥0) {I : Lp ℝ 2 P} |
| (hI : Tendsto (uniformSumLp hB hAd hind H hH hH2 T) atTop (𝓝 I)) : |
| (∫ ω, (I ω) ^ 2 ∂P) = |
| ∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, (H t.toNNReal ω) ^ 2 ∂P := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have henergy := EnergyRiemann.tendsto_energy_sum |
| (fun t => (hH2 t).aestronglyMeasurable) hc hC hb T |
| have hnorm := hI.norm.pow 2 |
| have henergy' : Tendsto |
| (fun n => ((T : ℝ) / n) * ∑ i ∈ Finset.range n, |
| ∫ ω, (H (uniformGrid T n i) ω) ^ 2 ∂P) atTop (𝓝 (‖I‖ ^ 2)) := by |
| apply hnorm.congr' |
| exact Filter.Eventually.of_forall (uniformSumLp_norm_sq hB hAd hind H hH hH2 T) |
| rw [← norm_sq_L2_eq_integral] |
| exact tendsto_nhds_unique henergy' henergy |
| |
| end Problem367.BrownianFiltrationGrid |
| |
| end |
| |
| end OAI |
| |
