| import OAI.Probability.FreeIsing.Brownian.SimpleIntegral |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367.BrownianFiltration |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ} |
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| theorem increment_indep_of_past |
| {s t : ℝ≥0} |
| (hind : Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) |
| {H : Ω → ℝ} (hH : StronglyMeasurable[F s] H) : |
| IndepFun (fun ω => B t ω - B s ω) H P := |
| indep_of_indep_of_le_right hind hH.measurable.comap_le |
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| theorem increment_condExp_zero |
| (hB : IsPreBrownianReal B P) (hAd : StronglyAdapted F B) |
| {s t : ℝ≥0} |
| (hind : Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) : |
| P[(fun ω => B t ω - B s ω) | F s] =ᵐ[P] fun _ => 0 := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hm : Measurable (fun ω => B t ω - B s ω) := |
| ((hAd t).mono (F.le t)).measurable.sub ((hAd s).mono (F.le s)).measurable |
| have hce := condExp_indep_eq hm.comap_le (F.le s) |
| (show StronglyMeasurable[MeasurableSpace.comap |
| (fun ω => B t ω - B s ω) inferInstance] (fun ω => B t ω - B s ω) from |
| (measurable_iff_comap_le.mpr le_rfl).stronglyMeasurable) hind |
| have hmean : (∫ ω, B t ω - B s ω ∂P) = 0 := by |
| rw [integral_sub (hB.integrable_eval t) (hB.integrable_eval s), |
| hB.integral_eval, hB.integral_eval, sub_self] |
| simpa only [hmean] using hce |
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| theorem increment_sq_condExp |
| (hB : IsPreBrownianReal B P) (hAd : StronglyAdapted F B) |
| {s t : ℝ≥0} (hst : s ≤ t) |
| (hind : Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) : |
| P[(fun ω => (B t ω - B s ω) ^ 2) | F s] =ᵐ[P] |
| fun _ => (t : ℝ) - (s : ℝ) := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hm : Measurable (fun ω => B t ω - B s ω) := |
| ((hAd t).mono (F.le t)).measurable.sub ((hAd s).mono (F.le s)).measurable |
| have hd : StronglyMeasurable[MeasurableSpace.comap |
| (fun ω => B t ω - B s ω) inferInstance] (fun ω => B t ω - B s ω) := |
| (measurable_iff_comap_le.mpr le_rfl).stronglyMeasurable |
| have hce := condExp_indep_eq hm.comap_le (F.le s) (hd.pow 2) hind |
| have hmean : (∫ ω, (B t ω - B s ω) ^ 2 ∂P) = (t : ℝ) - (s : ℝ) := by |
| have h := preBrownian_integral_sq (hB.shift s) (t - s) |
| simpa [add_tsub_cancel_of_le hst, NNReal.coe_sub hst] using h |
| change P[(fun ω => (B t ω - B s ω) ^ 2) | F s] =ᵐ[P] |
| fun _ => ∫ ω, (B t ω - B s ω) ^ 2 ∂P at hce |
| simpa only [hmean] using hce |
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| theorem predictable_mul_memLp_two |
| (hB : IsPreBrownianReal B P) {s t : ℝ≥0} |
| (hind : Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) |
| {H : Ω → ℝ} (hH : StronglyMeasurable[F s] H) (hH2 : MemLp H 2 P) : |
| MemLp (fun ω => H ω * (B t ω - B s ω)) 2 P := by |
| have hD2 : MemLp (fun ω => B t ω - B s ω) 2 P := |
| (hB.isGaussianProcess.hasGaussianLaw_eval t).memLp_two.sub |
| (hB.isGaussianProcess.hasGaussianLaw_eval s).memLp_two |
| have hi := (increment_indep_of_past hind hH).symm |
| have hint := (hi.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)).integrable_mul |
| hH2.integrable_sq hD2.integrable_sq |
| apply (memLp_two_iff_integrable_sq (hH2.aestronglyMeasurable.mul hD2.aestronglyMeasurable)).2 |
| simpa only [Function.comp_def, Pi.mul_def, mul_pow, id_eq] using hint |
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| theorem predictable_sum_isometry |
| (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) |
| (t : ℕ → ℝ≥0) (ht : Monotone t) (H : ℕ → Ω → ℝ) |
| (hH : ∀ i, StronglyMeasurable[F (t i)] (H i)) |
| (hH2 : ∀ i, MemLp (H i) 2 P) (s : Finset ℕ) : |
| (∫ ω, (∑ i ∈ s, H i ω * (B (t (i + 1)) ω - B (t i) ω)) ^ 2 ∂P) = |
| ∑ i ∈ s, ((t (i + 1) : ℝ) - (t i : ℝ)) * ∫ ω, H i ω ^ 2 ∂P := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| let G : Filtration ℕ mΩ := |
| ⟨fun i => F (t i), fun _ _ hij => F.mono (ht hij), fun i => F.le (t i)⟩ |
| let D : ℕ → Ω → ℝ := fun i ω => B (t (i + 1)) ω - B (t i) ω |
| have hD (i : ℕ) : StronglyMeasurable[G (i + 1)] (D i) := |
| (hAd (t (i + 1))).sub ((hAd (t i)).mono (F.mono (ht (Nat.le_succ i)))) |
| have hD2 (i : ℕ) : MemLp (D i) 2 P := |
| (hB.isGaussianProcess.hasGaussianLaw_eval (t (i + 1))).memLp_two.sub |
| (hB.isGaussianProcess.hasGaussianLaw_eval (t i)).memLp_two |
| exact SimpleIntegral.predictable_sum_isometry G H D |
| (fun i => (t (i + 1) : ℝ) - (t i : ℝ)) hH hD hD2 |
| (fun i => predictable_mul_memLp_two hB |
| (hind (t i) (t (i + 1)) (ht (Nat.le_succ i))) (hH i) (hH2 i)) |
| (fun i => increment_condExp_zero hB hAd |
| (hind (t i) (t (i + 1)) (ht (Nat.le_succ i)))) |
| (fun i => increment_sq_condExp hB hAd (ht (Nat.le_succ i)) |
| (hind (t i) (t (i + 1)) (ht (Nat.le_succ i)))) s |
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| theorem increments_indep_natural |
| (hB : IsPreBrownianReal B P) (hm : ∀ t, StronglyMeasurable (B t)) |
| {s t : ℝ≥0} (hst : s ≤ t) : |
| Indep (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) |
| (Filtration.natural B hm s) P := by |
| rw [Filtration.natural_eq_comap] |
| exact preBrownian_increment_indep_past hB hst |
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| end Problem367.BrownianFiltration |
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| end |
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| end OAI |
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