| import OAI.Probability.FreeIsing.Brownian.FiltrationIntegralProperties | |
| import OAI.Probability.FreeIsing.Stochastic.StoppedBrownianIntegral | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped NNReal Topology | |
| namespace Problem367.BrownianIntegralProcess | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} | |
| theorem meanSquare_condExp_sub_le [IsFiniteMeasure P] | |
| (m : MeasurableSpace Ω) {X Y : Ω → ℝ} | |
| (hX : MemLp X 2 P) (hY : MemLp Y 2 P) : | |
| (∫ ω, (P[X | m] ω - P[Y | m] ω) ^ 2 ∂P) ≤ | |
| ∫ ω, (X ω - Y ω) ^ 2 ∂P := by | |
| have hsub := condExp_sub (hX.integrable (by norm_num)) | |
| (hY.integrable (by norm_num)) m | |
| calc | |
| _ = ∫ ω, (P[X - Y | m] ω) ^ 2 ∂P := by | |
| apply integral_congr_ae | |
| filter_upwards [hsub] with ω hω | |
| change P[X - Y | m] ω = P[X | m] ω - P[Y | m] ω at hω | |
| rw [hω] | |
| _ ≤ _ := by | |
| have hbound := integral_norm_condExp_rpow_le (m := m) | |
| (by norm_num : (1 : ℝ) ≤ 2) | |
| (f := X - Y) (by | |
| simpa only [Real.rpow_two, Real.norm_eq_abs, sq_abs, Pi.sub_apply] using | |
| (hX.sub hY).integrable_sq) | |
| simpa only [Real.rpow_two, Real.norm_eq_abs, sq_abs, Pi.sub_apply] using hbound | |
| theorem finite_stoppedSum_eq_condExp [IsFiniteMeasure P] | |
| {F : Filtration ℝ≥0 mΩ} {B : ℝ≥0 → Ω → ℝ} | |
| (hB : Martingale B F P) (s : Finset ℕ) (a b : ℕ → ℝ≥0) | |
| (H : ℕ → Ω → ℝ) (hab : ∀ i ∈ s, a i ≤ b i) | |
| (hH : ∀ i ∈ s, StronglyMeasurable[F (a i)] (H i)) | |
| (C : ℕ → ℝ) (hb : ∀ i ∈ s, ∀ᵐ ω ∂P, ‖H i ω‖ ≤ C i) (t : ℝ≥0) : | |
| P[(fun ω => ∑ i ∈ s, H i ω * (B (b i) ω - B (a i) ω)) | F t] =ᵐ[P] | |
| fun ω => ∑ i ∈ s, StoppedSimpleIntegral.intervalSum B (H i) (a i) (b i) t ω := by | |
| have hint (i : ℕ) (hi : i ∈ s) : | |
| Integrable (fun ω => H i ω * (B (b i) ω - B (a i) ω)) P := | |
| ((hB.integrable (b i)).sub (hB.integrable (a i))).bdd_mul | |
| ((hH i hi).mono (F.le (a i))).aestronglyMeasurable (hb i hi) | |
| have hsum := condExp_finsetSum hint (F t) | |
| have hsumFun : (∑ i ∈ s, fun ω => H i ω * (B (b i) ω - B (a i) ω)) = | |
| (fun ω => ∑ i ∈ s, H i ω * (B (b i) ω - B (a i) ω)) := by | |
| funext ω | |
| simp only [Finset.sum_apply] | |
| rw [hsumFun] at hsum | |
| have heach : ∀ᵐ ω ∂P, ∀ i ∈ s, | |
| P[(fun ω => H i ω * (B (b i) ω - B (a i) ω)) | F t] ω = | |
| StoppedSimpleIntegral.intervalSum B (H i) (a i) (b i) t ω := | |
| s.eventually_all.mpr (fun i hi => | |
| StoppedSimpleIntegral.intervalSum_eq_condExp_terminal hB (hab i hi) (hH i hi) (hb i hi) t) | |
| filter_upwards [hsum, heach] with ω hsum heach | |
| simp only [Finset.sum_apply] at hsum | |
| rw [hsum] | |
| exact Finset.sum_congr rfl (fun i hi => heach i hi) | |
| def stoppedUniformSum (B H : ℝ≥0 → Ω → ℝ) (T : ℝ≥0) (n : ℕ) | |
| (t : ℝ≥0) (ω : Ω) : ℝ := | |
| ∑ i ∈ Finset.range n, StoppedSimpleIntegral.intervalSum B | |
| (H (BrownianGrid.uniformGrid T n i)) (BrownianGrid.uniformGrid T n i) | |
| (BrownianGrid.uniformGrid T n (i + 1)) t ω | |
| theorem stoppedUniformSum_eq_condExp | |
| {B H : 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) | |
| (hH : StronglyAdapted F H) {C : ℝ} | |
| (hb : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C) (T : ℝ≥0) (n : ℕ) (t : ℝ≥0) : | |
| P[BrownianGrid.uniformSum B H T n | F t] =ᵐ[P] stoppedUniformSum B H T n t := by | |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure | |
| apply finite_stoppedSum_eq_condExp (BrownianFiltration.martingale_of_independent_past hB hAd hind) | |
| (Finset.range n) (BrownianGrid.uniformGrid T n) | |
| (fun i => BrownianGrid.uniformGrid T n (i + 1)) | |
| (fun i => H (BrownianGrid.uniformGrid T n i)) | |
| (fun i _ => BrownianGrid.uniformGrid_mono T n (Nat.le_succ i)) | |
| (fun i _ => hH _) (fun _ => C) | |
| intro i hi | |
| filter_upwards [hb] with ω hω | |
| simpa only [Real.norm_eq_abs] using hω (BrownianGrid.uniformGrid T n i) | |
| theorem exists_stopped_uniform_integral_process | |
| {B H : 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) | |
| (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) : | |
| ∃ M : ℝ≥0 → Ω → ℝ, Martingale M F P ∧ | |
| (∀ t, MemLp (M t) 2 P) ∧ | |
| (∀ t, (∫ ω, M t ω ∂P) = 0) ∧ | |
| (∀ t, T ≤ t → M t = M T) ∧ | |
| (∫ ω, (M T ω) ^ 2 ∂P) = | |
| (∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, (H t.toNNReal ω) ^ 2 ∂P) ∧ | |
| (∀ t, Tendsto (fun n => ∫ ω, | |
| (stoppedUniformSum B H T n t ω - M t ω) ^ 2 ∂P) atTop (𝓝 0)) := by | |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure | |
| obtain ⟨I, hIm, hI2, hIzero, hIiso, hIlim⟩ := | |
| BrownianFiltrationGrid.exists_terminal_uniform_integral_isometry | |
| hB hAd hind H hH hc hC hb T | |
| let M : ℝ≥0 → Ω → ℝ := fun t => P[I | F t] | |
| have hIint : Integrable I P := hI2.integrable (by norm_num) | |
| have hafter (t : ℝ≥0) (ht : T ≤ t) : M t = I := | |
| condExp_of_stronglyMeasurable (F.le t) (hIm.mono (F.mono ht)) hIint | |
| have hH2 (t : ℝ≥0) : MemLp (H t) 2 P := by | |
| apply MemLp.of_bound ((hH t).mono (F.le t)).aestronglyMeasurable C | |
| filter_upwards [hb] with ω hω | |
| simpa only [Real.norm_eq_abs] using hω t | |
| refine ⟨M, martingale_condExp I F P, ?_, ?_, ?_, ?_, ?_⟩ | |
| · intro t | |
| exact hI2.condExp (by norm_num) | |
| · intro t | |
| exact (integral_condExp (F.le t)).trans hIzero | |
| · intro t ht | |
| exact (hafter t ht).trans (hafter T le_rfl).symm | |
| · rw [hafter T le_rfl] | |
| exact hIiso | |
| · intro t | |
| apply squeeze_zero (fun n => integral_nonneg (fun ω => sq_nonneg _)) ?_ hIlim | |
| intro n | |
| calc | |
| _ = ∫ ω, (P[BrownianGrid.uniformSum B H T n | F t] ω - P[I | F t] ω) ^ 2 ∂P := by | |
| apply integral_congr_ae | |
| filter_upwards [stoppedUniformSum_eq_condExp hB hAd hind hH hb T n t] with ω hω | |
| change (stoppedUniformSum B H T n t ω - P[I | F t] ω) ^ 2 = _ | |
| rw [hω] | |
| _ ≤ _ := meanSquare_condExp_sub_le (F t) | |
| (BrownianFiltrationGrid.uniformSum_memLp hB hAd hind H hH hH2 T n) hI2 | |
| end Problem367.BrownianIntegralProcess | |
| end | |
| end OAI | |