| import Mathlib |
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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 |
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| theorem preBrownian_coordinate_of_gaussianArray |
| {ι Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
| {B : ι → ℝ≥0 → Ω → ℝ} |
| (hG : IsGaussianProcess (fun p : (_ : ι) × ℝ≥0 => B p.1 p.2) P) |
| (hmean : ∀ i t, ∫ ω, B i t ω ∂P = 0) |
| (hcov : ∀ i s t, s ≤ t → cov[B i s, B i t; P] = (s : ℝ)) |
| (i : ι) : IsPreBrownianReal (B i) P := by |
| have hGi : IsGaussianProcess (B i) P := hG.comp_right (Sigma.mk i) |
| exact hGi.isPreBrownianReal_of_covariance (hmean i) (hcov i) |
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| theorem iIndepFun_of_gaussianArray_crossCovariance |
| {ι Ω T : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
| {B : ι → T → Ω → ℝ} |
| (hG : IsGaussianProcess (fun p : (_ : ι) × T => B p.1 p.2) P) |
| (hcross : ∀ i j, i ≠ j → ∀ s t, cov[B i s, B j t; P] = 0) : |
| iIndepFun (fun i ω t => B i t ω) P := by |
| exact hG.iIndepFun_of_covariance_eq_zero |
| (fun i t => hG.aemeasurable ⟨i, t⟩) hcross |
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| theorem map_gaussianArray_eq_infinitePi |
| {ι Ω T : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
| {B : ι → T → Ω → ℝ} |
| (hG : IsGaussianProcess (fun p : (_ : ι) × T => B p.1 p.2) P) |
| (hm : ∀ i t, Measurable (B i t)) |
| (hcross : ∀ i j, i ≠ j → ∀ s t, cov[B i s, B j t; P] = 0) : |
| P.map (fun ω i t => B i t ω) = |
| Measure.infinitePi (fun i => P.map (fun ω t => B i t ω)) := by |
| exact (iIndepFun_of_gaussianArray_crossCovariance hG hcross).map_fun_eq_infinitePi_map |
| (fun i => Measurable.of_eval (hm i)) |
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| theorem map_preBrownian_process_eq |
| {Ω Ω' : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} {Q : Measure Ω'} |
| {B : ℝ≥0 → Ω → ℝ} {C : ℝ≥0 → Ω' → ℝ} |
| (hB : IsPreBrownianReal B P) (hC : IsPreBrownianReal C Q) |
| (hmB : ∀ t, Measurable (B t)) (hmC : ∀ t, Measurable (C t)) : |
| P.map (fun ω t => B t ω) = Q.map (fun ω t => C t ω) := by |
| have : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have : IsProbabilityMeasure Q := hC.isGaussianProcess.isProbabilityMeasure |
| have hp : IsProjectiveLimit (P.map (fun ω t => B t ω)) |
| ProbabilityTheory.BrownianReal.projectiveFamily := by |
| intro I |
| rw [Measure.map_map (Finset.measurable_restrict I) |
| (Measurable.of_eval hmB)] |
| exact (hB.hasLaw I).map_eq |
| have hq : IsProjectiveLimit (Q.map (fun ω t => C t ω)) |
| ProbabilityTheory.BrownianReal.projectiveFamily := by |
| intro I |
| rw [Measure.map_map (Finset.measurable_restrict I) |
| (Measurable.of_eval hmC)] |
| exact (hC.hasLaw I).map_eq |
| exact hp.unique hq |
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| theorem map_brownianArray_eq_infinitePi_reference |
| {ι Ω Ω' : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} {Q : Measure Ω'} |
| {B : ι → ℝ≥0 → Ω → ℝ} {C : ℝ≥0 → Ω' → ℝ} |
| (hG : IsGaussianProcess (fun p : (_ : ι) × ℝ≥0 => B p.1 p.2) P) |
| (hmB : ∀ i t, Measurable (B i t)) |
| (hmean : ∀ i t, ∫ ω, B i t ω ∂P = 0) |
| (hcov : ∀ i s t, s ≤ t → cov[B i s, B i t; P] = (s : ℝ)) |
| (hcross : ∀ i j, i ≠ j → ∀ s t, cov[B i s, B j t; P] = 0) |
| (hC : IsPreBrownianReal C Q) (hmC : ∀ t, Measurable (C t)) : |
| P.map (fun ω i t => B i t ω) = |
| Measure.infinitePi (fun _ : ι => Q.map (fun ω t => C t ω)) := by |
| rw [map_gaussianArray_eq_infinitePi hG hmB hcross] |
| congr 1 |
| funext i |
| exact map_preBrownian_process_eq |
| (preBrownian_coordinate_of_gaussianArray hG hmean hcov i) hC (hmB i) hmC |
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| theorem iIndepFun_of_measurableEmbedding |
| {ι Ω E F : Type*} [MeasurableSpace Ω] [MeasurableSpace E] |
| [MeasurableSpace F] {P : Measure Ω} {X : ι → Ω → E} |
| {e : E → F} (he : MeasurableEmbedding e) |
| (h : iIndepFun (fun i => e ∘ X i) P) : iIndepFun X P := by |
| rw [iIndepFun_iff_iIndep] at h ⊢ |
| simpa only [← MeasurableSpace.comap_comp, he.comap_eq] using h |
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| section ContinuousPaths |
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| variable {T : Type*} [TopologicalSpace T] [TopologicalSpace.SeparableSpace T] |
| [Nonempty T] [MeasurableSpace C(T, ℝ)] [BorelSpace C(T, ℝ)] |
| [StandardBorelSpace C(T, ℝ)] |
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| theorem measurableEmbedding_continuousPath_denseSeq : |
| MeasurableEmbedding |
| (fun (f : C(T, ℝ)) n => f (TopologicalSpace.denseSeq T n)) := by |
| apply Measurable.measurableEmbedding |
| · exact Measurable.of_eval (fun _ => (continuous_eval_const _).measurable) |
| · intro f g h |
| apply ContinuousMap.coe_injective |
| exact (TopologicalSpace.denseRange_denseSeq T).equalizer f.continuous g.continuous h |
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| theorem measurable_continuousPath_of_evaluations |
| {Ω : Type*} [MeasurableSpace Ω] {X : Ω → C(T, ℝ)} |
| (hm : ∀ t, Measurable (fun ω => X ω t)) : Measurable X := by |
| apply measurableEmbedding_continuousPath_denseSeq.measurable_comp_iff.mp |
| exact Measurable.of_eval (fun index => hm (TopologicalSpace.denseSeq T index)) |
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| theorem iIndepFun_continuousPaths_of_evaluations |
| {ι Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
| {X : ι → Ω → C(T, ℝ)} |
| (h : iIndepFun (fun i ω t => X i ω t) P) : iIndepFun X P := by |
| apply iIndepFun_of_measurableEmbedding measurableEmbedding_continuousPath_denseSeq |
| exact h.comp (fun _ f n => f (TopologicalSpace.denseSeq T n)) |
| (fun _ => Measurable.of_eval (fun _ => measurable_pi_apply _)) |
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| theorem map_continuousPaths_eq_of_evaluations |
| {Ω Ω' : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} {Q : Measure Ω'} |
| {X : Ω → C(T, ℝ)} {Y : Ω' → C(T, ℝ)} |
| (hmX : Measurable X) (hmY : Measurable Y) |
| (h : P.map (fun ω t => X ω t) = Q.map (fun ω t => Y ω t)) : |
| P.map X = Q.map Y := by |
| apply measurableEmbedding_continuousPath_denseSeq.map_injective |
| rw [Measure.map_map measurableEmbedding_continuousPath_denseSeq.measurable hmX, |
| Measure.map_map measurableEmbedding_continuousPath_denseSeq.measurable hmY] |
| have hs : Measurable (fun f : T → ℝ => fun n => f (TopologicalSpace.denseSeq T n)) := |
| Measurable.of_eval (fun _ => measurable_pi_apply _) |
| have hX : Measurable (fun ω t => X ω t) := |
| Measurable.of_eval (fun time => (continuous_eval_const time).measurable.comp hmX) |
| have hY : Measurable (fun ω t => Y ω t) := |
| Measurable.of_eval (fun time => (continuous_eval_const time).measurable.comp hmY) |
| have hh := congrArg |
| (Measure.map (fun f : T → ℝ => fun n => f (TopologicalSpace.denseSeq T n))) h |
| simpa only [Measure.map_map hs hX, Measure.map_map hs hY, Function.comp_def] using hh |
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| end ContinuousPaths |
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| section ContinuousBrownianPaths |
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| variable [MeasurableSpace C(ℝ≥0, ℝ)] [BorelSpace C(ℝ≥0, ℝ)] |
| [StandardBorelSpace C(ℝ≥0, ℝ)] |
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| theorem map_continuousPreBrownian_eq |
| {Ω Ω' : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} {Q : Measure Ω'} |
| {X : Ω → C(ℝ≥0, ℝ)} {Y : Ω' → C(ℝ≥0, ℝ)} |
| (hmX : Measurable X) (hmY : Measurable Y) |
| (hX : IsPreBrownianReal (fun t ω => X ω t) P) |
| (hY : IsPreBrownianReal (fun t ω => Y ω t) Q) : |
| P.map X = Q.map Y := by |
| apply map_continuousPaths_eq_of_evaluations hmX hmY |
| exact map_preBrownian_process_eq hX hY |
| (fun t => (continuous_eval_const t).measurable.comp hmX) |
| (fun t => (continuous_eval_const t).measurable.comp hmY) |
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| theorem map_continuousBrownianArray_eq_infinitePi_reference |
| {ι Ω Ω' : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} {Q : Measure Ω'} |
| {X : ι → Ω → C(ℝ≥0, ℝ)} {Y : Ω' → C(ℝ≥0, ℝ)} |
| (hmX : ∀ i, Measurable (X i)) (hmY : Measurable Y) |
| (hG : IsGaussianProcess (fun p : (_ : ι) × ℝ≥0 => fun ω => X p.1 ω p.2) P) |
| (hmean : ∀ i t, ∫ ω, X i ω t ∂P = 0) |
| (hcov : ∀ i s t, s ≤ t → cov[fun ω => X i ω s, fun ω => X i ω t; P] = (s : ℝ)) |
| (hcross : ∀ i j, i ≠ j → ∀ s t, |
| cov[fun ω => X i ω s, fun ω => X j ω t; P] = 0) |
| (hY : IsPreBrownianReal (fun t ω => Y ω t) Q) : |
| P.map (fun ω i => X i ω) = |
| Measure.infinitePi (fun _ : ι => Q.map Y) := by |
| have hi : iIndepFun X P := iIndepFun_continuousPaths_of_evaluations |
| (iIndepFun_of_gaussianArray_crossCovariance hG hcross) |
| rw [hi.map_fun_eq_infinitePi_map hmX] |
| congr 1 |
| funext i |
| exact map_continuousPreBrownian_eq (hmX i) hmY |
| (preBrownian_coordinate_of_gaussianArray hG hmean hcov i) hY |
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| end ContinuousBrownianPaths |
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| end Problem367 |
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| end |
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| end OAI |
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