| import OAI.Probability.FreeIsing.Brownian.IntegralProcess |
| import OAI.Probability.FreeIsing.Estimates.CompensatedSquareLimit |
| import OAI.Probability.FreeIsing.Stochastic.SelectedCompensator |
| import OAI.Probability.FreeIsing.Stochastic.StoppedBrownianSquare |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal Topology |
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| namespace Problem367.BrownianIntegralProcess |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
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| theorem stopped_uniform_limit_compensatedSquare |
| {B H M : 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) (hM : StronglyAdapted F M) (hM2 : ∀ t, MemLp (M t) 2 P) |
| (hlim : ∀ t, Tendsto (fun n => ∫ ω, |
| (stoppedUniformSum B H T n t ω - M t ω) ^ 2 ∂P) atTop (𝓝 0)) : |
| Martingale (fun t ω => (M t ω) ^ 2 - SelectedCompensator.selected H T t ω) F P := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hH2 (t : ℝ≥0) : MemLp (H t) 2 P := by |
| apply MemLp.of_bound hH.stronglyMeasurable.aestronglyMeasurable C |
| filter_upwards [hb] with ω hω |
| simpa only [Real.norm_eq_abs] using hω t |
| have hS2 (n : ℕ) (t : ℝ≥0) : MemLp (stoppedUniformSum B H T n t) 2 P := |
| BrownianFiltration.stopped_predictable_sum_memLp_two hB hind |
| (BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n) |
| (fun i => H (BrownianGrid.uniformGrid T n i)) (fun i => hH _) (fun i => hH2 _) |
| (Finset.range n) t |
| have hfinite (n : ℕ) : Martingale (fun t ω => |
| (stoppedUniformSum B H T n t ω) ^ 2 - SelectedCompensator.squareSum H T n t ω) F P := by |
| have h := StoppedBrownianSquare.stopped_predictable_sum_square_martingale hB hAd hind |
| (BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n) |
| (fun i => H (BrownianGrid.uniformGrid T n i)) (fun i => hH _) (fun _ => C) |
| (fun i => hb.mono (fun ω hω => by simpa only [Real.norm_eq_abs] using hω _)) n |
| simpa only [stoppedUniformSum, SelectedCompensator.squareSum, mul_comm] using h |
| apply CompensatedSquareLimit.martingale_compensatedSquare_of_limits (l := atTop) |
| hS2 (fun n t => (SelectedCompensator.squareSum_memLp hH hC hb T t n 1).integrable le_rfl) |
| hfinite hM hM2 (SelectedCompensator.selected_stronglyAdapted hH T) |
| (fun t => (SelectedCompensator.selected_memLp hH hc hC hb T t 1).integrable le_rfl) |
| · intro t |
| exact CompensatedSquareLimit.tendsto_eLpNorm_two_of_meanSquare |
| (fun n => hS2 n t) (hM2 t) (hlim t) |
| · intro t |
| exact SelectedCompensator.tendsto_squareSum_L1_selected |
| (fun s => hH.stronglyMeasurable.aestronglyMeasurable) hc hC hb T t |
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| theorem exists_stopped_uniform_integral_compensatedSquare |
| {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 Q : ℝ≥0 → Ω → ℝ, Martingale M F P ∧ |
| (∀ t, MemLp (M t) 2 P) ∧ StronglyAdapted F Q ∧ |
| (∀ᵐ ω ∂P, Continuous (fun t => Q t ω)) ∧ |
| (∀ᵐ ω ∂P, ∀ t, Q t ω = |
| ∫ s in (0 : ℝ)..min (t : ℝ) T, (H s.toNNReal ω) ^ 2) ∧ |
| Martingale (fun t ω => (M t ω) ^ 2 - Q t ω) F P ∧ |
| (∀ t, Tendsto (fun n => ∫ ω, |
| (stoppedUniformSum B H T n t ω - M t ω) ^ 2 ∂P) atTop (𝓝 0)) := by |
| obtain ⟨M, hM, hM2, _, _, _, hlim⟩ := |
| exists_stopped_uniform_integral_process hB hAd hind hH hc hC hb T |
| exact ⟨M, SelectedCompensator.selected H T, hM, hM2, |
| SelectedCompensator.selected_stronglyAdapted hH T, |
| SelectedCompensator.ae_continuous_selected hc T, |
| SelectedCompensator.ae_selected_eq_integral hc T, |
| stopped_uniform_limit_compensatedSquare hB hAd hind hH hc hC hb T hM.stronglyAdapted hM2 hlim, |
| hlim⟩ |
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| end Problem367.BrownianIntegralProcess |
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| end |
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| end OAI |
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