| import OAI.Probability.FreeIsing.Martingales.ContinuousSelection | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped NNReal Topology | |
| namespace Problem367.MartingaleSquareMaximal | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] | |
| omit [IsFiniteMeasure μ] in | |
| theorem dist_toLp_sq {f g : Ω → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : | |
| dist (hf.toLp f) (hg.toLp g) ^ 2 = ∫ ω, (f ω - g ω) ^ 2 ∂μ := by | |
| rw [dist_eq_norm, ← real_inner_self_eq_norm_sq, L2.inner_def] | |
| simp only [real_inner_self_eq_norm_sq, Real.norm_eq_abs, sq_abs] | |
| apply integral_congr_ae | |
| filter_upwards [Lp.coeFn_sub (hf.toLp f) (hg.toLp g), hf.coeFn_toLp, | |
| hg.coeFn_toLp] with ω hsub ha hb | |
| simp only [Pi.sub_apply] at hsub | |
| rw [hsub, ha, hb] | |
| theorem dist_toLp_le_of_le {F : Filtration ℝ≥0 mΩ} {M N : ℝ≥0 → Ω → ℝ} | |
| (hM : Martingale M F μ) (hN : Martingale N F μ) | |
| (hM2 : ∀ t, MemLp (M t) 2 μ) (hN2 : ∀ t, MemLp (N t) 2 μ) | |
| {s t : ℝ≥0} (hst : s ≤ t) : | |
| dist ((hM2 s).toLp (M s)) ((hN2 s).toLp (N s)) ≤ | |
| dist ((hM2 t).toLp (M t)) ((hN2 t).toLp (N t)) := by | |
| have hsq := integral_sq_le (hM.sub hN) (fun r => (hM2 r).sub (hN2 r)) hst | |
| simp only [Pi.sub_apply] at hsq | |
| rw [← dist_toLp_sq (hM2 s) (hN2 s), ← dist_toLp_sq (hM2 t) (hN2 t)] at hsq | |
| exact (sq_le_sq₀ dist_nonneg dist_nonneg).mp hsq | |
| theorem L2_cauchy_of_le {F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ} | |
| (hM : ∀ n, Martingale (M n) F μ) (h2 : ∀ n t, MemLp (M n t) 2 μ) | |
| {s T : ℝ≥0} (hs : s ≤ T) | |
| (hC : CauchySeq (fun n => (h2 n T).toLp (M n T))) : | |
| CauchySeq (fun n => (h2 n s).toLp (M n s)) := by | |
| apply Metric.cauchySeq_iff.mpr | |
| intro ε hε | |
| obtain ⟨N, hN⟩ := Metric.cauchySeq_iff.mp hC ε hε | |
| refine ⟨N, fun n hn m hm => ?_⟩ | |
| exact (dist_toLp_le_of_le (hM n) (hM m) (h2 n) (h2 m) hs).trans_lt | |
| (hN n hn m hm) | |
| theorem memLp_tendsto_of_cauchy_of_ae_subsequence | |
| {X : ℕ → Ω → ℝ} (h2 : ∀ n, MemLp (X n) 2 μ) {L : Ω → ℝ} | |
| (hC : CauchySeq (fun n => (h2 n).toLp (X n))) | |
| {v : ℕ → ℕ} (hv : Tendsto v atTop atTop) | |
| (hL : ∀ᵐ ω ∂μ, Tendsto (fun n => X (v n) ω) atTop (𝓝 (L ω))) : | |
| ∃ hL2 : MemLp L 2 μ, | |
| Tendsto (fun n => (h2 n).toLp (X n)) atTop (𝓝 (hL2.toLp L)) := by | |
| obtain ⟨I, hI⟩ := cauchySeq_tendsto_of_complete hC | |
| have hraw : TendstoInMeasure μ X atTop I := | |
| (tendstoInMeasure_of_tendsto_Lp hI).congr_left (fun n => (h2 n).coeFn_toLp) | |
| have heq : L =ᵐ[μ] I := tendstoInMeasure_ae_unique | |
| (tendstoInMeasure_of_tendsto_ae (fun n => (h2 (v n)).aestronglyMeasurable) hL) | |
| (hraw.comp hv) | |
| have hL2 : MemLp L 2 μ := (Lp.memLp I).ae_eq heq.symm | |
| refine ⟨hL2, ?_⟩ | |
| have hclass : hL2.toLp L = I := | |
| (hL2.toLp_congr (Lp.memLp I) heq).trans (Lp.toLp_coeFn I (Lp.memLp I)) | |
| rw [hclass] | |
| exact hI | |
| theorem exists_predictable_continuous_L2_limit | |
| {F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ} | |
| (hM : ∀ n, Martingale (M n) F μ) | |
| (h2 : ∀ n t, MemLp (M n t) 2 μ) | |
| (hc : ∀ n, ∀ᵐ ω ∂μ, Continuous (fun t => M n t ω)) (T : ℝ≥0) | |
| (hC : CauchySeq (fun n => (h2 n T).toLp (M n T))) : | |
| ∃ L : ℝ≥0 → Ω → ℝ, IsStronglyPredictable F L ∧ | |
| (∀ᵐ ω ∂μ, ContinuousOn (fun t => L t ω) (Iic T)) ∧ | |
| ∀ t ≤ T, ∃ hL2 : MemLp (L t) 2 μ, | |
| Tendsto (fun n => (h2 n t).toLp (M n t)) atTop (𝓝 (hL2.toLp (L t))) := by | |
| obtain ⟨v, L, hv, hpred, hcont, hlim⟩ := | |
| exists_predictable_continuous_subsequence_of_terminal_L2_cauchy hM h2 hc T hC | |
| refine ⟨L, hpred, hcont, fun t ht => ?_⟩ | |
| apply memLp_tendsto_of_cauchy_of_ae_subsequence (fun n => h2 n t) | |
| (L2_cauchy_of_le hM h2 ht hC) hv.tendsto_atTop | |
| filter_upwards [hlim] with ω hω | |
| exact hω t ht | |
| end Problem367.MartingaleSquareMaximal | |
| end | |
| end OAI | |