| import OAI.Probability.FreeIsing.Conditioning.Characteristic |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
| open Problem367.ConditionalCharacteristic |
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| namespace Problem367.BrownianCharacterization |
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| variable {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] |
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| theorem condExp_character_of_setIntegral |
| (hm : m ≤ mΩ) {X : Ω → ℝ} (hX : Measurable X) (c : ℂ) (u : ℝ) |
| (hchar : ∀ A : Set Ω, MeasurableSet[m] A → |
| (∫ ω in A, character X u ω ∂μ) = μ.real A * c) : |
| μ[character X u | m] =ᵐ[μ] fun _ => c := by |
| symm |
| apply ae_eq_condExp_of_forall_setIntegral_eq hm (integrable_character hX u) |
| · intro A _ _ |
| exact (integrable_const c).integrableOn |
| · intro A hA _ |
| rw [hchar A hA] |
| simp |
| · exact stronglyMeasurable_const.aestronglyMeasurable |
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| theorem hasIndepIncrements_of_past_events_general |
| {T E : Type*} [Preorder T] [MeasurableSpace E] [Sub E] [MeasurableSub₂ E] |
| {B : T → Ω → E} (F : Filtration T mΩ) (hB : Adapted F B) |
| (hind : ∀ s t : T, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A → |
| ∀ S : Set E, MeasurableSet S → |
| μ (A ∩ (fun ω => B t ω - B s ω) ⁻¹' S) = |
| μ A * μ ((fun ω => B t ω - B s ω) ⁻¹' S)) : |
| HasIndepIncrements B μ := by |
| intro n t ht |
| rw [iIndepFun_iff_measure_inter_preimage_eq_mul] |
| intro S sets |
| induction S using Finset.induction_on_max with |
| | empty => simp |
| | insert a S hlt ih => |
| intro hsets |
| have ha : a ∉ S := by |
| intro ha |
| exact lt_irrefl a (hlt a ha) |
| have hmsets : ∀ i ∈ S, MeasurableSet (sets i) := |
| fun i hi => hsets i (Finset.mem_insert_of_mem hi) |
| have hA : MeasurableSet[F (t a.castSucc)] |
| (⋂ i ∈ S, (fun ω => B (t i.succ) ω - B (t i.castSucc) ω) ⁻¹' sets i) := by |
| apply S.measurableSet_biInter |
| intro i hi |
| have hi1 : t i.succ ≤ t a.castSucc := ht (by |
| change i.val + 1 ≤ a.val |
| exact Nat.succ_le_of_lt (hlt i hi)) |
| have hi0 : t i.castSucc ≤ t a.castSucc := ht (by |
| change i.val ≤ a.val |
| exact (hlt i hi).le) |
| exact ((hB.measurable_le hi1).sub (hB.measurable_le hi0)) (hmsets i hi) |
| have h := hind (t a.castSucc) (t a.succ) (ht (by change a.val ≤ a.val + 1; omega)) _ hA |
| (sets a) (hsets a (Finset.mem_insert_self a S)) |
| rw [Finset.prod_insert ha] |
| rw [Finset.set_biInter_insert] |
| rw [Set.inter_comm, h, ih hmsets, mul_comm] |
|
|
| theorem hasIndepIncrements_of_past_events |
| {B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hB : StronglyAdapted F B) |
| (hind : ∀ s t : ℝ≥0, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A → |
| ∀ S : Set ℝ, MeasurableSet S → |
| μ (A ∩ (fun ω => B t ω - B s ω) ⁻¹' S) = |
| μ A * μ ((fun ω => B t ω - B s ω) ⁻¹' S)) : |
| HasIndepIncrements B μ := |
| hasIndepIncrements_of_past_events_general F hB.adapted hind |
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| theorem hasIndepIncrements_of_conditional_increment_laws |
| {T E : Type*} [Preorder T] [MeasurableSpace E] [Sub E] [MeasurableSub₂ E] |
| {B : T → Ω → E} (F : Filtration T mΩ) (hB : Adapted F B) |
| (ν : T → T → Measure E) |
| (hlaw : ∀ s t : T, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A → |
| (μ.restrict A).map (fun ω => B t ω - B s ω) = μ A • ν s t) : |
| HasIndepIncrements B μ := by |
| apply hasIndepIncrements_of_past_events_general F hB |
| intro s t hst A hA S hS |
| have hm : Measurable (fun ω => B t ω - B s ω) := |
| hB.measurable.sub hB.measurable |
| have huniv : μ.map (fun ω => B t ω - B s ω) = ν s t := by |
| simpa using hlaw s t hst Set.univ MeasurableSet.univ |
| have h := congrArg (fun κ : Measure E => κ S) (hlaw s t hst A hA) |
| rw [Measure.map_apply hm hS, Measure.restrict_apply (hm hS), |
| Measure.smul_apply, smul_eq_mul, ← huniv, Measure.map_apply hm hS] at h |
| simpa only [Set.inter_comm] using h |
|
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| theorem isPreBrownianReal_of_conditional_charFun |
| {B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hB : StronglyAdapted F B) (hzero : ∀ᵐ ω ∂μ, B 0 ω = 0) |
| (hchar : ∀ s t : ℝ≥0, s ≤ t → ∀ u : ℝ, |
| μ[character (fun ω => B t ω - B s ω) u | F s] |
| =ᵐ[μ] fun _ => charFun (gaussianReal 0 (t - s)) u) : |
| IsPreBrownianReal B μ := by |
| have hm (t : ℝ≥0) : Measurable (B t) := ((hB t).mono (F.le t)).measurable |
| have hlaw (s t : ℝ≥0) (hst : s ≤ t) : |
| HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s)) μ := |
| hasLaw_of_condExp_character (F.le s) ((hm t).sub (hm s)) _ (hchar s t hst) |
| apply HasIndepIncrements.isPreBrownianReal_of_hasLaw |
| · intro t |
| have h := hlaw 0 t (by positivity) |
| simp only [tsub_zero] at h |
| apply h.congr |
| filter_upwards [hzero] with ω hω |
| simp [hω] |
| · apply hasIndepIncrements_of_past_events F hB |
| intro s t hst A hA S hS |
| have hmap := map_restrict_eq_smul_of_condExp_character (F.le s) |
| ((hm t).sub (hm s)) (gaussianReal 0 (t - s)) (hchar s t hst) hA |
| have h := congrArg (fun κ : Measure ℝ => κ S) hmap |
| rw [Measure.map_apply ((hm t).sub (hm s)) hS, |
| Measure.restrict_apply (((hm t).sub (hm s)) hS), |
| Measure.smul_apply, smul_eq_mul] at h |
| have hmass : μ ((fun ω => B t ω - B s ω) ⁻¹' S) = gaussianReal 0 (t - s) S := by |
| rw [← (hlaw s t hst).map_eq] |
| exact (Measure.map_apply (show Measurable (fun ω => B t ω - B s ω) from |
| (hm t).sub (hm s)) hS).symm |
| rw [hmass, Set.inter_comm] |
| exact h |
|
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| theorem isPreBrownianReal_of_conditional_exponential |
| {B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hB : StronglyAdapted F B) (hzero : ∀ᵐ ω ∂μ, B 0 ω = 0) |
| (hchar : ∀ s t : ℝ≥0, s ≤ t → ∀ u : ℝ, |
| μ[character (fun ω => B t ω - B s ω) u | F s] =ᵐ[μ] |
| fun _ => Complex.exp (-(((t - s : ℝ≥0) : ℝ) : ℂ) * (u : ℂ)^2 / 2)) : |
| IsPreBrownianReal B μ := by |
| apply isPreBrownianReal_of_conditional_charFun F hB hzero |
| intro s t hst u |
| simpa only [charFun_gaussianReal, Complex.ofReal_zero, mul_zero, zero_mul, |
| zero_sub, neg_mul, neg_div] using hchar s t hst u |
|
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| theorem isPreBrownianReal_of_setIntegral_character |
| {B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hB : StronglyAdapted F B) (hzero : ∀ᵐ ω ∂μ, B 0 ω = 0) |
| (hchar : ∀ s t : ℝ≥0, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A → |
| ∀ u : ℝ, (∫ ω in A, character (fun ω => B t ω - B s ω) u ω ∂μ) = |
| μ.real A * Complex.exp (-(((t - s : ℝ≥0) : ℝ) : ℂ) * (u : ℂ)^2 / 2)) : |
| IsPreBrownianReal B μ := by |
| apply isPreBrownianReal_of_conditional_exponential F hB hzero |
| intro s t hst u |
| exact condExp_character_of_setIntegral (F.le s) |
| (((hB t).mono (F.le t)).measurable.sub ((hB s).mono (F.le s)).measurable) |
| _ u (fun A hA => hchar s t hst A hA u) |
|
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| theorem isBrownianReal_of_setIntegral_character |
| {B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ) |
| (hB : StronglyAdapted F B) (hzero : ∀ᵐ ω ∂μ, B 0 ω = 0) |
| (hcontinuous : ∀ᵐ ω ∂μ, Continuous (fun t => B t ω)) |
| (hchar : ∀ s t : ℝ≥0, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A → |
| ∀ u : ℝ, (∫ ω in A, character (fun ω => B t ω - B s ω) u ω ∂μ) = |
| μ.real A * Complex.exp (-(((t - s : ℝ≥0) : ℝ) : ℂ) * (u : ℂ)^2 / 2)) : |
| IsBrownianReal B μ := |
| ⟨isPreBrownianReal_of_setIntegral_character F hB hzero hchar, hcontinuous⟩ |
|
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| end Problem367.BrownianCharacterization |
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| end |
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| end OAI |
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