| import OAI.Probability.FreeIsing.Brownian.ContinuousIntegral |
| import OAI.Probability.FreeIsing.Martingales.RandomEvaluation |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal 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_random_evaluation |
| {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) |
| (hBc : ∀ᵐ ω ∂P, Continuous (fun t => B t ω)) |
| (hH : StronglyAdapted F H) |
| {C : ℝ} (hb : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C) |
| {T : ℝ≥0} {I : ℝ≥0 → Ω → ℝ} |
| (hI : Martingale I F P) (hI2 : ∀ t, MemLp (I t) 2 P) |
| (hIc : ∀ᵐ ω ∂P, Continuous (fun t => I t ω)) |
| (hlim : Tendsto (fun n => ∫ ω, |
| (stoppedUniformSum B H T n T ω - I T ω) ^ 2 ∂P) atTop (𝓝 0)) |
| {τ : Ω → ℝ≥0} (hτ : Measurable τ) (hτT : ∀ ω, τ ω ≤ T) : |
| Tendsto (fun n => ∫ ω, |
| (stoppedUniformSum B H T n (τ ω) ω - I (τ ω) ω) ^ 2 ∂P) atTop (𝓝 0) := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hH2 (t : ℝ≥0) : MemLp (H t) 2 P := by |
| apply MemLp.of_bound ((hH t).mono (F.le t)).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 hSM (n : ℕ) : Martingale (stoppedUniformSum B H T n) F P := by |
| apply BrownianFiltration.stopped_predictable_sum_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 => ?_) (Finset.range n) |
| filter_upwards [hb] with ω hω |
| simpa only [Real.norm_eq_abs] using hω (BrownianGrid.uniformGrid T n i) |
| exact MartingaleRandomEvaluation.tendsto_integral_sq_random_evaluation_sub |
| hSM hI hS2 hI2 (ae_continuous_stoppedUniformSum hBc T) hIc hτ hτT hlim |
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| end Problem367.BrownianIntegralProcess |
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| end |
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| end OAI |
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