| import OAI.Probability.FreeIsing.Brownian.UniformIntegral |
| import OAI.Probability.FreeIsing.Factors.ContinuousRecovery |
| import OAI.Probability.FreeIsing.Spectral.KoopmanInvolutions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
|
|
| namespace Problem367.BrownianGrid |
|
|
| theorem uniformSum_stronglyMeasurable_terminal |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, StronglyMeasurable (B t)) |
| (H : ℝ≥0 → Ω → ℝ) (hH : StronglyAdapted (Filtration.natural B hm) H) |
| (T : ℝ≥0) (n : ℕ) : |
| StronglyMeasurable[Filtration.natural B hm T] (uniformSum B H T n) := by |
| by_cases hn : n = 0 |
| · subst n |
| have hz : uniformSum B H T 0 = fun _ : Ω => (0 : ℝ) := by |
| funext ω |
| simp [uniformSum, integralSum] |
| rw [hz] |
| exact stronglyMeasurable_const |
| have hn0 : 0 < n := Nat.pos_of_ne_zero hn |
| unfold uniformSum integralSum |
| apply Finset.stronglyMeasurable_fun_sum |
| intro i hi |
| have hi0 : i ≤ n := (Finset.mem_range.mp hi).le |
| have hi1 : i + 1 ≤ n := Finset.mem_range.mp hi |
| have hti := uniformGrid_le_horizon T hn0 hi0 |
| have hti1 := uniformGrid_le_horizon T hn0 hi1 |
| exact ((hH _).mono ((Filtration.natural B hm).mono hti)).mul |
| (((Filtration.stronglyAdapted_natural hm _).mono ((Filtration.natural B hm).mono hti1)).sub |
| ((Filtration.stronglyAdapted_natural hm _).mono ((Filtration.natural B hm).mono hti))) |
|
|
| theorem integral_integralSum_eq_zero |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hm : ∀ t, StronglyMeasurable (B t)) (t : ℕ → ℝ≥0) (ht : Monotone t) |
| (H : ℕ → Ω → ℝ) |
| (hH : ∀ i, StronglyMeasurable[Filtration.natural B hm (t i)] (H i)) |
| (hH2 : ∀ i, MemLp (H i) 2 P) (n : ℕ) : |
| (∫ ω, integralSum B t H n ω ∂P) = 0 := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hD2 (i : ℕ) : MemLp (fun ω => B (t (i + 1)) ω - B (t i) ω) 2 P := |
| (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two.sub |
| (hB.isGaussianProcess.hasGaussianLaw_eval _).memLp_two |
| have hint (i : ℕ) : Integrable (fun ω => H i ω * (B (t (i + 1)) ω - B (t i) ω)) P := |
| (hH2 i).integrable_mul (hD2 i) |
| unfold integralSum |
| rw [integral_finsetSum _ (fun i _ => hint i)] |
| apply Finset.sum_eq_zero |
| intro i hi |
| exact SimpleIntegral.integral_mul_eq_zero_of_condExp_eq_zero |
| ((Filtration.natural B hm).le _) (hH i) ((hD2 i).integrable (by norm_num)) (hint i) |
| (preBrownian_increment_condExp_zero hB hm (ht (Nat.le_succ i))) |
|
|
| theorem integral_sq_error_tendsto_of_Lp_tendsto |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hm : ∀ t, StronglyMeasurable (B t)) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (Filtration.natural B hm) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) (T : ℝ≥0) {I : Lp ℝ 2 P} |
| (hI : Tendsto (uniformSumLp hB hm H hH hH2 T) atTop (𝓝 I)) : |
| Tendsto (fun n => ∫ ω, (uniformSum B H T n ω - I ω) ^ 2 ∂P) atTop (𝓝 0) := by |
| 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 hm H hH hH2 T n) I, |
| coe_uniformSumLp hB hm H hH hH2 T n] with ω hsub hsum |
| simp only [Pi.sub_apply] at hsub |
| rw [hsub, hsum] |
| |
| theorem uniformSumLp_limit_terminal_centered |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hm : ∀ t, StronglyMeasurable (B t)) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (Filtration.natural B hm) H) |
| (hH2 : ∀ t, MemLp (H t) 2 P) (T : ℝ≥0) {I : Lp ℝ 2 P} |
| (hI : Tendsto (uniformSumLp hB hm H hH hH2 T) atTop (𝓝 I)) : |
| AEStronglyMeasurable[Filtration.natural B hm T] (I : Ω → ℝ) P ∧ |
| (∫ ω, I ω ∂P) = 0 := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| constructor |
| · apply (isClosed_aestronglyMeasurable ((Filtration.natural B hm).le T)).mem_of_tendsto hI |
| apply Filter.Eventually.of_forall |
| intro n |
| exact (uniformSum_stronglyMeasurable_terminal hm H hH T n).aestronglyMeasurable.congr |
| (coe_uniformSumLp hB hm H hH hH2 T n).symm |
| · have hinner := ((innerSL ℝ (Lp.const 2 P (1 : ℝ))).continuous.tendsto I).comp hI |
| have hzero (n : ℕ) : inner ℝ (Lp.const 2 P (1 : ℝ)) (uniformSumLp hB hm H hH hH2 T n) = 0 := by |
| rw [inner_const_one_eq_integral] |
| rw [integral_congr_ae (coe_uniformSumLp hB hm H hH hH2 T n)] |
| exact integral_integralSum_eq_zero hB hm (uniformGrid T n) (uniformGrid_mono T n) |
| (fun i => H (uniformGrid T n i)) (fun i => hH _) (fun i => hH2 _) n |
| change Tendsto (fun n => inner ℝ (Lp.const 2 P (1 : ℝ)) |
| (uniformSumLp hB hm H hH hH2 T n)) atTop |
| (𝓝 (inner ℝ (Lp.const 2 P (1 : ℝ)) I)) at hinner |
| have heq : inner ℝ (Lp.const 2 P (1 : ℝ)) I = 0 := by |
| apply tendsto_nhds_unique hinner |
| simpa only [hzero] using |
| (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0)) |
| simpa only [inner_const_one_eq_integral] using heq |
| |
| theorem exists_terminal_uniform_integral |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hm : ∀ t, StronglyMeasurable (B t)) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (Filtration.natural B hm) 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[Filtration.natural B hm T] I ∧ MemLp I 2 P ∧ |
| (∫ ω, I ω ∂P) = 0 ∧ |
| 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 hm H hH hH2 T hcont |
| obtain ⟨hIm, hIzero⟩ := uniformSumLp_limit_terminal_centered hB hm H hH hH2 T hI |
| refine ⟨hIm.mk I, hIm.stronglyMeasurable_mk, ?_, ?_, ?_⟩ |
| · exact MemLp.ae_eq hIm.ae_eq_mk (Lp.memLp I) |
| · exact (integral_congr_ae hIm.ae_eq_mk).symm.trans hIzero |
| · apply (integral_sq_error_tendsto_of_Lp_tendsto hB hm H hH hH2 T hI).congr' |
| apply Filter.Eventually.of_forall |
| intro n |
| apply integral_congr_ae |
| filter_upwards [hIm.ae_eq_mk] with ω hω |
| rw [hω] |
|
|
| theorem ae_eq_of_mean_square_limits |
| {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
| {X : ℕ → Ω → ℝ} {I J : Ω → ℝ} |
| (hX : ∀ n, MemLp (X n) 2 P) (hI : MemLp I 2 P) (hJ : MemLp J 2 P) |
| (hlimI : Tendsto (fun n => ∫ ω, (X n ω - I ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (hlimJ : Tendsto (fun n => ∫ ω, (X n ω - J ω) ^ 2 ∂P) atTop (𝓝 0)) : |
| I =ᵐ[P] J := by |
| have hb (n : ℕ) : (∫ ω, (I ω - J ω) ^ 2 ∂P) ≤ |
| 2 * (∫ ω, (X n ω - I ω) ^ 2 ∂P) + 2 * (∫ ω, (X n ω - J ω) ^ 2 ∂P) := by |
| simpa only [sub_sq_comm] using integral_sq_sub_le_two hI hJ (hX n) |
| have hz : (∫ ω, (I ω - J ω) ^ 2 ∂P) ≤ 0 := by |
| apply le_of_tendsto_of_tendsto tendsto_const_nhds |
| (show Tendsto (fun n => 2 * (∫ ω, (X n ω - I ω) ^ 2 ∂P) + |
| 2 * (∫ ω, (X n ω - J ω) ^ 2 ∂P)) atTop (𝓝 0) from ?_) |
| (Filter.Eventually.of_forall hb) |
| simpa only [mul_zero, add_zero] using (hlimI.const_mul 2).add (hlimJ.const_mul 2) |
| exact ae_eq_of_integral_sq_sub_eq_zero (hI.sub hJ).integrable_sq |
| (le_antisymm hz (integral_nonneg (fun ω => sq_nonneg _))) |
|
|
| theorem exists_terminal_uniform_integral_of_bounded_continuous |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hm : ∀ t, StronglyMeasurable (B t)) (H : ℝ≥0 → Ω → ℝ) |
| (hH : StronglyAdapted (Filtration.natural B hm) 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[Filtration.natural B hm T] I ∧ MemLp I 2 P ∧ |
| (∫ ω, I ω ∂P) = 0 ∧ |
| 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 ((Filtration.natural B hm).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_terminal_uniform_integral hB hm H hH hH2 T |
| intro ε hε |
| exact MeanSquareContinuity.uniform_integral_sq_sub_le hHm hc hC hb T hε |
|
|
| end Problem367.BrownianGrid |
|
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| end |
|
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| end OAI |
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