| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter |
| open scoped NNReal ENNReal |
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| namespace Problem367.MartingaleSquareMaximal |
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| variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Preorder ι] |
| {μ : Measure Ω} {F : Filtration ι mΩ} {M : ι → Ω → ℝ} |
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| theorem submartingale_sq (hM : Martingale M F μ) |
| (h2 : ∀ t, MemLp (M t) 2 μ) : |
| Submartingale (fun t ω => M t ω ^ 2) F μ := by |
| refine ⟨fun t => (hM.stronglyMeasurable t).pow 2, ?_, fun t => (h2 t).integrable_sq⟩ |
| intro s t hst |
| have hint : Integrable (fun ω => ‖M t ω‖ ^ (2 : ℝ)) μ := by |
| simpa only [Real.rpow_two, Real.norm_eq_abs, sq_abs] using (h2 t).integrable_sq |
| have hj := Integrable.norm_condExp_rpow_le (m := F s) (by norm_num : (1 : ℝ) ≤ 2) hint |
| have hj' : (fun ω => μ[M t | F s] ω ^ 2) ≤ᵐ[μ] |
| μ[(fun ω => M t ω ^ 2) | F s] := by |
| simpa only [Real.rpow_two, Real.norm_eq_abs, sq_abs] using hj |
| filter_upwards [hj', hM.condExp_ae_eq hst] with ω hω heq |
| simpa only [heq] using hω |
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| theorem finite_maximal_sq [IsFiniteMeasure μ] |
| {F : Filtration ℕ mΩ} {M : ℕ → Ω → ℝ} |
| (hM : Martingale M F μ) (h2 : ∀ t, MemLp (M t) 2 μ) |
| (ε : ℝ≥0) (n : ℕ) : |
| (ε : ℝ≥0∞) ^ 2 * μ {ω | ∃ k ≤ n, (ε : ℝ) ≤ |M k ω|} ≤ |
| ENNReal.ofReal (∫ ω, M n ω ^ 2 ∂μ) := by |
| have hs := submartingale_sq hM h2 |
| have hmax := maximal_ineq hs (fun t ω => sq_nonneg (M t ω)) (ε := ε ^ 2) n |
| have hevent : {ω | ((ε ^ 2 : ℝ≥0) : ℝ) ≤ |
| (Finset.range (n + 1)).sup' Finset.nonempty_range_add_one (fun k => M k ω ^ 2)} = |
| {ω | ∃ k ≤ n, (ε : ℝ) ≤ |M k ω|} := by |
| ext ω |
| simp only [Set.mem_ofPred_eq, Finset.le_sup'_iff, Finset.mem_range, Nat.lt_succ_iff, |
| NNReal.coe_pow] |
| constructor |
| · rintro ⟨k, hk, hval⟩ |
| refine ⟨k, hk, ?_⟩ |
| nlinarith [abs_nonneg (M k ω), sq_abs (M k ω), ε.coe_nonneg] |
| · rintro ⟨k, hk, hval⟩ |
| refine ⟨k, hk, ?_⟩ |
| nlinarith [abs_nonneg (M k ω), sq_abs (M k ω), ε.coe_nonneg] |
| rw [hevent] at hmax |
| calc |
| _ ≤ ENNReal.ofReal (∫ ω in {ω | ∃ k ≤ n, (ε : ℝ) ≤ |M k ω|}, M n ω ^ 2 ∂μ) := by |
| simpa only [ENNReal.coe_pow] using hmax |
| _ ≤ _ := ENNReal.ofReal_le_ofReal (setIntegral_le_integral (h2 n).integrable_sq |
| (Filter.Eventually.of_forall (fun ω => sq_nonneg (M n ω)))) |
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| theorem integral_sq_le [IsFiniteMeasure μ] |
| (hM : Martingale M F μ) (h2 : ∀ t, MemLp (M t) 2 μ) |
| {s t : ι} (hst : s ≤ t) : |
| (∫ ω, M s ω ^ 2 ∂μ) ≤ ∫ ω, M t ω ^ 2 ∂μ := by |
| have hs := submartingale_sq hM h2 |
| calc |
| _ ≤ ∫ ω, μ[(fun ω => M t ω ^ 2) | F s] ω ∂μ := |
| integral_mono_ae (h2 s).integrable_sq integrable_condExp (hs.2.1 s t hst) |
| _ = _ := integral_condExp (F.le s) |
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| theorem sampled_maximal_sq [IsFiniteMeasure μ] |
| (hM : Martingale M F μ) (h2 : ∀ t, MemLp (M t) 2 μ) |
| (q : ℕ → ι) (hq : Monotone q) (ε : ℝ≥0) (n : ℕ) |
| {T : ι} (hT : q n ≤ T) : |
| (ε : ℝ≥0∞) ^ 2 * μ {ω | ∃ k ≤ n, (ε : ℝ) ≤ |M (q k) ω|} ≤ |
| ENNReal.ofReal (∫ ω, M T ω ^ 2 ∂μ) := by |
| let G : Filtration ℕ mΩ := ⟨fun k => F (q k), |
| fun _ _ h => F.mono (hq h), fun k => F.le (q k)⟩ |
| have hMq : Martingale (fun k => M (q k)) G μ := |
| ⟨fun k => hM.stronglyMeasurable (q k), fun i j hij => hM.condExp_ae_eq (hq hij)⟩ |
| exact (finite_maximal_sq hMq (fun k => h2 (q k)) ε n).trans |
| (ENNReal.ofReal_le_ofReal (integral_sq_le hM h2 hT)) |
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| end Problem367.MartingaleSquareMaximal |
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| end |
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| end OAI |
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