| import OAI.Probability.FreeIsing.Martingales.ContinuousSelection |
| import OAI.Probability.FreeIsing.Martingales.StrongMaximal |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped NNReal Topology |
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| namespace Problem367.MartingaleRandomEvaluation |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
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| theorem stronglyMeasurable_random_evaluation |
| {F : Filtration ℝ≥0 mΩ} {M : ℝ≥0 → Ω → ℝ} |
| (hp : IsStronglyProgressive F M) {τ : Ω → ℝ≥0} (hτ : Measurable τ) |
| {T : ℝ≥0} (hbound : ∀ ω, τ ω ≤ T) : |
| StronglyMeasurable (fun ω => M (τ ω) ω) := by |
| have hm : StronglyMeasurable (fun p : Iic T × Ω => M p.1 p.2) := |
| (hp T).mono (sup_le_sup le_rfl (MeasurableSpace.comap_mono (F.le T))) |
| let τ' : Ω → Iic T := fun ω => ⟨τ ω, hbound ω⟩ |
| have ht : Measurable τ' := hτ.subtype_mk |
| exact hm.comp_measurable (ht.prodMk measurable_id) |
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|
| theorem aestronglyMeasurable_random_evaluation |
| {F : Filtration ℝ≥0 mΩ} {M : ℝ≥0 → Ω → ℝ} |
| (hM : StronglyAdapted F M) |
| (hc : ∀ᵐ ω ∂P, Continuous (fun t => M t ω)) |
| {τ : Ω → ℝ≥0} (hτ : Measurable τ) {T : ℝ≥0} (hbound : ∀ ω, τ ω ≤ T) : |
| AEStronglyMeasurable (fun ω => M (τ ω) ω) P := by |
| have hm := stronglyMeasurable_random_evaluation |
| (PredictableMesh.repair_isStronglyPredictable hM).isStronglyProgressive hτ hbound |
| apply hm.aestronglyMeasurable.congr |
| filter_upwards [hc] with ω hω |
| exact PredictableMesh.repair_eq_of_continuousOn (T := τ ω) hω.continuousOn le_rfl |
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| theorem integrable_sq_random_evaluation [IsFiniteMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {M : ℝ≥0 → Ω → ℝ} |
| (hM : Martingale M F P) (h2 : ∀ t, MemLp (M t) 2 P) |
| (hc : ∀ᵐ ω ∂P, Continuous (fun t => M t ω)) |
| {τ : Ω → ℝ≥0} (hτ : Measurable τ) {T : ℝ≥0} (hbound : ∀ ω, τ ω ≤ T) : |
| Integrable (fun ω => M (τ ω) ω ^ 2) P := by |
| apply (MartingaleStrongMaximal.integrable_maximalSquare hM h2 hc T).mono' |
| ((aestronglyMeasurable_random_evaluation hM.stronglyAdapted hc hτ hbound).pow 2) |
| filter_upwards [MartingaleStrongMaximal.ae_all_sq_le_maximalSquare hM h2 hc T] with ω hω |
| change ‖M (τ ω) ω ^ 2‖ ≤ _ |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg (M (τ ω) ω))] using |
| hω (τ ω) (hbound ω) |
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| theorem memLp_random_evaluation [IsFiniteMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {M : ℝ≥0 → Ω → ℝ} |
| (hM : Martingale M F P) (h2 : ∀ t, MemLp (M t) 2 P) |
| (hc : ∀ᵐ ω ∂P, Continuous (fun t => M t ω)) |
| {τ : Ω → ℝ≥0} (hτ : Measurable τ) {T : ℝ≥0} (hbound : ∀ ω, τ ω ≤ T) : |
| MemLp (fun ω => M (τ ω) ω) 2 P := |
| (memLp_two_iff_integrable_sq |
| (aestronglyMeasurable_random_evaluation hM.stronglyAdapted hc hτ hbound)).mpr |
| (integrable_sq_random_evaluation hM h2 hc hτ hbound) |
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| theorem integral_sq_random_evaluation_le [IsFiniteMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {M : ℝ≥0 → Ω → ℝ} |
| (hM : Martingale M F P) (h2 : ∀ t, MemLp (M t) 2 P) |
| (hc : ∀ᵐ ω ∂P, Continuous (fun t => M t ω)) |
| {τ : Ω → ℝ≥0} (hτ : Measurable τ) {T : ℝ≥0} (hbound : ∀ ω, τ ω ≤ T) : |
| (∫ ω, M (τ ω) ω ^ 2 ∂P) ≤ 4 * ∫ ω, M T ω ^ 2 ∂P := by |
| apply le_trans (integral_mono_ae |
| (integrable_sq_random_evaluation hM h2 hc hτ hbound) |
| (MartingaleStrongMaximal.integrable_maximalSquare hM h2 hc T) ?_) |
| (MartingaleStrongMaximal.integral_maximalSquare_le hM h2 hc T) |
| filter_upwards [MartingaleStrongMaximal.ae_all_sq_le_maximalSquare hM h2 hc T] with ω hω |
| exact hω (τ ω) (hbound ω) |
|
|
| theorem tendsto_integral_sq_random_evaluation_sub [IsFiniteMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ} {L : ℝ≥0 → Ω → ℝ} |
| (hM : ∀ n, Martingale (M n) F P) (hL : Martingale L F P) |
| (h2 : ∀ n t, MemLp (M n t) 2 P) (hL2 : ∀ t, MemLp (L t) 2 P) |
| (hc : ∀ n, ∀ᵐ ω ∂P, Continuous (fun t => M n t ω)) |
| (hLc : ∀ᵐ ω ∂P, Continuous (fun t => L t ω)) |
| {τ : Ω → ℝ≥0} (hτ : Measurable τ) {T : ℝ≥0} (hbound : ∀ ω, τ ω ≤ T) |
| (hlim : Tendsto (fun n => ∫ ω, (M n T ω - L T ω) ^ 2 ∂P) atTop (𝓝 0)) : |
| Tendsto (fun n => ∫ ω, (M n (τ ω) ω - L (τ ω) ω) ^ 2 ∂P) atTop (𝓝 0) := by |
| have hdiff (n : ℕ) : ∀ᵐ ω ∂P, Continuous (fun t => (M n - L) t ω) := by |
| filter_upwards [hc n, hLc] with ω hm hl |
| exact hm.sub hl |
| apply squeeze_zero (fun n => integral_nonneg (fun ω => sq_nonneg _)) |
| (fun n => ?_) (by simpa only [mul_zero] using hlim.const_mul 4) |
| simpa only [Pi.sub_apply] using integral_sq_random_evaluation_le |
| ((hM n).sub hL) (fun t => (h2 n t).sub (hL2 t)) (hdiff n) hτ hbound |
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| end Problem367.MartingaleRandomEvaluation |
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| end |
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| end OAI |
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