| import OAI.Probability.FreeIsing.Brownian.ArrayFiltrationIntegral |
| import OAI.Probability.FreeIsing.Brownian.IntegralProcess |
| import OAI.Probability.FreeIsing.Estimates.CompensatedSquareLimit |
| import OAI.Probability.FreeIsing.Stochastic.SelectedCompensator |
| import OAI.Probability.FreeIsing.Stochastic.StoppedBrownianArraySquare |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal Topology BigOperators |
| |
| namespace Problem367.BrownianArrayCompensator |
| |
| variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι] |
| {P : Measure Ω} [IsProbabilityMeasure P] {F : Filtration ℝ≥0 mΩ} |
|
|
| theorem compensatedSquare_of_terminal_meanSquare |
| {B H : ι → ℝ≥0 → Ω → ℝ} {M : ℝ≥0 → Ω → ℝ} |
| (hB : ∀ j, IsPreBrownianReal (B j) P) |
| (hAd : ∀ j, StronglyAdapted F (B j)) |
| (hpast : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω j => B j t ω - B j s ω) inferInstance) (F s) P) |
| (hcross : ∀ s t, s ≤ t → Pairwise fun j k => |
| IndepFun (fun ω => B j t ω - B j s ω) (fun ω => B k t ω - B k s ω) P) |
| (hH : ∀ j, StronglyAdapted F (H j)) |
| (hc : ∀ j, ∀ᵐ ω ∂P, Continuous (fun t => H j t ω)) |
| (C : ι → ℝ) (hC : ∀ j, 0 ≤ C j) |
| (hb : ∀ j, ∀ᵐ ω ∂P, ∀ t, |H j t ω| ≤ C j) |
| (hM : Martingale M F P) (hM2 : ∀ t, MemLp (M t) 2 P) |
| (hlim : ∀ T, Tendsto (fun n : ℕ => ∫ ω, |
| ((∑ j, BrownianGrid.uniformSum (B j) (H j) T n ω) - M T ω) ^ 2 ∂P) |
| atTop (𝓝 0)) : |
| Martingale (fun t ω => M t ω ^ 2 - |
| ∑ j, SelectedCompensator.selected (H j) t t ω) F P := by |
| classical |
| have hi (j : ι) (s t : ℝ≥0) (hst : s ≤ t) := |
| BrownianArrayFiltration.coordinate_increment_indep_past (hpast s t hst) j |
| have hH2 (j : ι) (t : ℝ≥0) : MemLp (H j t) 2 P := by |
| apply MemLp.of_bound (hH j).stronglyMeasurable.aestronglyMeasurable (C j) |
| filter_upwards [hb j] with ω hω |
| simpa only [Real.norm_eq_abs] using hω t |
| have hU2 (T : ℝ≥0) (n : ℕ) (j : ι) : |
| MemLp (BrownianGrid.uniformSum (B j) (H j) T n) 2 P := |
| BrownianFiltrationGrid.uniformSum_memLp (hB j) (hAd j) (hi j) |
| (H j) (hH j) (hH2 j) T n |
| have hQ : StronglyAdapted F (fun t ω => |
| ∑ j, SelectedCompensator.selected (H j) t t ω) := by |
| intro t |
| exact Finset.stronglyMeasurable_fun_sum _ fun j _ => |
| SelectedCompensator.selected_stronglyAdapted (hH j) t t |
| refine ⟨fun t => ((hM.stronglyAdapted t).pow 2).sub (hQ t), ?_⟩ |
| intro s T hsT |
| let U (n : ℕ) : Ω → ℝ := fun ω => |
| ∑ j, BrownianGrid.uniformSum (B j) (H j) T n ω |
| let X (n : ℕ) (t : ℝ≥0) (ω : Ω) : ℝ := |
| ∑ j, BrownianIntegralProcess.stoppedUniformSum (B j) (H j) T n t ω |
| let A (n : ℕ) (t : ℝ≥0) (ω : Ω) : ℝ := |
| ∑ j, SelectedCompensator.squareSum (H j) T n t ω |
| let Y (t : ℝ≥0) : Ω → ℝ := P[M T | F t] |
| let Q (t : ℝ≥0) (ω : Ω) : ℝ := |
| ∑ j, SelectedCompensator.selected (H j) T t ω |
| have hu2 (n : ℕ) : MemLp (U n) 2 P := |
| memLp_finsetSum _ fun j _ => hU2 T n j |
| have hx2 (n : ℕ) (t : ℝ≥0) : MemLp (X n t) 2 P := by |
| apply memLp_finsetSum |
| intro j hj |
| exact BrownianFiltration.stopped_predictable_sum_memLp_two (hB j) (hi j) |
| (BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n) |
| (fun i => H j (BrownianGrid.uniformGrid T n i)) (fun i => hH j _) |
| (fun i => hH2 j _) (Finset.range n) t |
| have ha (n : ℕ) (t : ℝ≥0) : Integrable (A n t) P := |
| integrable_finsetSum _ fun j _ => |
| (SelectedCompensator.squareSum_memLp (hH j) (hC j) (hb j) T t n 1).integrable le_rfl |
| have hq (t : ℝ≥0) : Integrable (Q t) P := |
| integrable_finsetSum _ fun j _ => |
| (SelectedCompensator.selected_memLp (hH j) (hc j) (hC j) (hb j) T t 1).integrable le_rfl |
| have hy2 (t : ℝ≥0) : MemLp (Y t) 2 P := (hM2 T).condExp (by norm_num) |
| have hce (n : ℕ) (t : ℝ≥0) : P[U n | F t] =ᵐ[P] X n t := by |
| have hsum := condExp_finsetSum (s := Finset.univ) |
| (fun j _ => (hU2 T n j).integrable (by norm_num)) (F t) |
| have heach : ∀ᵐ ω ∂P, ∀ j, |
| P[BrownianGrid.uniformSum (B j) (H j) T n | F t] ω = |
| BrownianIntegralProcess.stoppedUniformSum (B j) (H j) T n t ω := |
| ae_all_iff.mpr fun j => BrownianIntegralProcess.stoppedUniformSum_eq_condExp |
| (hB j) (hAd j) (hi j) (hH j) (hb j) T n t |
| have hsumFun : (∑ j, BrownianGrid.uniformSum (B j) (H j) T n) = U n := by |
| funext ω |
| simp only [U, Finset.sum_apply] |
| rw [hsumFun] at hsum |
| filter_upwards [hsum, heach] with ω hω he |
| simp only [Finset.sum_apply] at hω |
| exact hω.trans (Finset.sum_congr rfl (fun j _ => he j)) |
| have hxlim (t : ℝ≥0) : Tendsto (fun n => ∫ ω, |
| (X n t ω - Y t ω) ^ 2 ∂P) atTop (𝓝 0) := by |
| apply squeeze_zero (fun n => integral_nonneg (fun ω => sq_nonneg _)) ?_ (hlim T) |
| intro n |
| calc |
| _ = ∫ ω, (P[U n | F t] ω - P[M T | F t] ω) ^ 2 ∂P := by |
| apply integral_congr_ae |
| filter_upwards [hce n t] with ω hω |
| exact congrArg (fun z => (z - Y t ω) ^ 2) hω.symm |
| _ ≤ _ := BrownianIntegralProcess.meanSquare_condExp_sub_le (F t) (hu2 n) (hM2 T) |
| have hfinite (n : ℕ) : Martingale (fun t ω => (X n t ω) ^ 2 - A n t ω) F P := by |
| have hh := StoppedBrownianArraySquare.stopped_brownian_array_square_martingale |
| B hB hAd hpast hcross (BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n) |
| (fun j i => H j (BrownianGrid.uniformGrid T n i)) (fun j i => hH j _) |
| (fun j _ => C j) |
| (fun j i => (hb j).mono (fun ω hω => by simpa only [Real.norm_eq_abs] using hω _)) |
| Finset.univ n |
| simpa only [X, A, BrownianIntegralProcess.stoppedUniformSum, |
| SelectedCompensator.squareSum, mul_comm] using hh |
| have halim (t : ℝ≥0) : Tendsto (fun n => eLpNorm (A n t - Q t) 1 P) atTop (𝓝 0) := by |
| have hl := tendsto_finsetSum Finset.univ (fun j _ => |
| SelectedCompensator.tendsto_squareSum_L1_selected |
| (fun r => (hH j).stronglyMeasurable.aestronglyMeasurable) |
| (hc j) (hC j) (hb j) T t) |
| simp only [Finset.sum_const_zero] at hl |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hl (fun _ => bot_le) |
| intro n |
| change eLpNorm (A n t - Q t) 1 P ≤ ∑ j, |
| eLpNorm (SelectedCompensator.squareSum (H j) T n t - SelectedCompensator.selected (H j) T t) 1 P |
| have heq : A n t - Q t = ∑ j, |
| (SelectedCompensator.squareSum (H j) T n t - SelectedCompensator.selected (H j) T t) := by |
| funext ω |
| simp only [A, Q, Pi.sub_apply, Finset.sum_apply, Finset.sum_sub_distrib] |
| rw [heq] |
| exact eLpNorm_sum_le le_rfl |
| have hcomp := CompensatedSquareLimit.martingale_compensatedSquare_of_limits |
| hx2 ha hfinite (martingale_condExp (M T) F P).stronglyAdapted hy2 |
| (fun t => Finset.stronglyMeasurable_fun_sum _ fun j _ => |
| SelectedCompensator.selected_stronglyAdapted (hH j) T t) hq |
| (fun t => CompensatedSquareLimit.tendsto_eLpNorm_two_of_meanSquare |
| (fun n => hx2 n t) (hy2 t) (hxlim t)) halim |
| have heq (r : ℝ≥0) (hr : r ≤ T) : |
| (fun ω => (Y r ω) ^ 2 - Q r ω) =ᵐ[P] |
| (fun ω => M r ω ^ 2 - ∑ j, SelectedCompensator.selected (H j) r r ω) := by |
| have hcAll : ∀ᵐ ω ∂P, ∀ j, Continuous (fun t => H j t ω) := ae_all_iff.mpr hc |
| filter_upwards [hM.condExp_ae_eq hr, hcAll] with ω hω hcω |
| change (P[M T | F r] ω) ^ 2 - _ = _ |
| rw [hω] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro j hj |
| simp only [SelectedCompensator.selected_eq_integral_of_continuous (hcω j), |
| min_self, min_eq_left (show (r : ℝ) ≤ T from hr)] |
| exact (condExp_congr_ae (heq T le_rfl).symm).trans |
| ((hcomp.condExp_ae_eq hsT).trans (heq s hsT)) |
|
|
| end Problem367.BrownianArrayCompensator |
| |
| end |
|
|
| end OAI |
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|
