| import OAI.Probability.FreeIsing.Brownian.ContinuousIntegral |
| import OAI.Probability.FreeIsing.Brownian.IntegralCompensator |
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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 exists_continuous_integral_with_compensator |
| {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) (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, |
| IsStronglyPredictable F M ∧ IsStronglyPredictable F Q ∧ |
| Martingale M F P ∧ (∀ t, MemLp (M t) 2 P) ∧ |
| (∀ᵐ ω ∂P, Continuous (fun t => M t ω)) ∧ M 0 =ᵐ[P] 0 ∧ |
| (∀ᵐ ω ∂P, Continuous (fun t => Q t ω) ∧ Monotone (fun t => Q t ω)) ∧ |
| Q 0 =ᵐ[P] 0 ∧ (∀ t, Integrable (Q t) P) ∧ |
| (∀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 ∧ |
| (∀ᵐ ω ∂P, ∀ s t, |Q t ω - Q s ω| ≤ C ^ 2 * |(t : ℝ) - s|) ∧ |
| (∀ t, Tendsto (fun n => ∫ ω, |
| (stoppedUniformSum B H T n t ω - M t ω) ^ 2 ∂P) atTop (𝓝 0)) := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| obtain ⟨M, hp, hM, hMc, hM2, hMzero, _, _, _, hlim⟩ := |
| exists_continuous_stopped_uniform_integral hB hAd hind hBc hH hc hC hb T |
| let Q := PredictableMesh.repair (SelectedCompensator.selected H T) |
| have hQp : IsStronglyPredictable F Q := |
| PredictableMesh.repair_isStronglyPredictable |
| (SelectedCompensator.selected_stronglyAdapted hH T) |
| have hQeq : ∀ᵐ ω ∂P, ∀ t, Q t ω = SelectedCompensator.selected H T t ω := by |
| filter_upwards [SelectedCompensator.ae_continuous_selected hc T] with ω hω |
| intro t |
| exact PredictableMesh.repair_eq_of_continuousOn (T := t) hω.continuousOn le_rfl |
| have hcomp := stopped_uniform_limit_compensatedSquare |
| hB hAd hind hH hc hC hb T hM.stronglyAdapted hM2 hlim |
| refine ⟨M, Q, hp, hQp, hM, hM2, hMc, hMzero, ?_, ?_, ?_, ?_, ?_, ?_, hlim⟩ |
| · filter_upwards [hQeq, SelectedCompensator.ae_continuous_selected hc T, hc] |
| with ω heq hcont hcω |
| constructor |
| · exact hcont.congr (fun t => (heq t).symm) |
| · intro s t hst |
| change Q s ω ≤ Q t ω |
| rw [heq s, heq t] |
| exact SelectedCompensator.selected_monotone_of_continuous hcω T hst |
| · filter_upwards [hQeq] with ω hω |
| simp only [hω 0, SelectedCompensator.selected_zero, Pi.zero_apply] |
| · intro t |
| apply ((SelectedCompensator.selected_memLp hH hc hC hb T t 1).integrable le_rfl).congr |
| filter_upwards [hQeq] with ω hω |
| exact (hω t).symm |
| · filter_upwards [hQeq, SelectedCompensator.ae_selected_eq_integral hc T] with ω heq hω |
| intro t |
| exact (heq t).trans (hω t) |
| · apply hcomp.congr (fun t => ((hM.stronglyAdapted t).pow 2).sub (hQp.stronglyAdapted t)) |
| intro t |
| filter_upwards [hQeq] with ω hω |
| change M t ω ^ 2 - SelectedCompensator.selected H T t ω = M t ω ^ 2 - Q t ω |
| rw [hω t] |
| · filter_upwards [hQeq, hc, hb] with ω heq hcω hbω |
| intro s t |
| rw [heq s, heq t] |
| exact SelectedCompensator.abs_selected_sub_le hcω hC hbω T s t |
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| end Problem367.BrownianIntegralProcess |
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| end |
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| end OAI |
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