openAI-math-dataset / lean /OAI /Probability /FreeIsing /Conditioning /ContinuousCopyIndependence.lean
| import OAI.Probability.FreeIsing.Conditioning.ConstructedContinuousCopies | |
| import OAI.Probability.FreeIsing.Conditioning.Copies | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped NNReal | |
| namespace Problem367.ContinuousCopyBrownian | |
| open WienerSampler ContinuousObservationHistory ContinuousNoiseSplit | |
| variable {ι : Type*} | |
| theorem measurable_commonNoise_joinedFiltration (t : ℝ≥0) : | |
| Measurable[joinedFiltration (ι := ι) t] | |
| (fun q : Joint (ι := ι) => fun i => q.1 i t) := by | |
| change Measurable[pastSpace (ι := ι) t] _ | |
| have hh : Measurable[pastSpace (ι := ι) t] (history (ι := ι) t) := | |
| comap_measurable _ | |
| let : MeasurableSpace (Joint (ι := ι)) := pastSpace (ι := ι) t | |
| apply Measurable.of_eval | |
| intro i | |
| have hm := (continuous_eval_const t).measurable.comp | |
| ((measurable_pi_apply i).comp (measurable_fst.comp hh)) | |
| simpa only [Function.comp_def, history, stoppedArray, stoppedPath_apply, min_self] using hm | |
| theorem indep_future_joinedFiltration [Countable ι] | |
| (rho : Measure (ι → Path)) [IsProbabilityMeasure rho] | |
| (eta : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel eta] | |
| (s : ℝ≥0) | |
| (hind : IndepFun (fun q : (ι → Path) × (ι → Path) => | |
| (stoppedArray s q.1, stoppedArray s q.2)) | |
| (fun q => futureArray s q.1) (rho ⊗ₘ eta)) : | |
| Indep (MeasurableSpace.comap | |
| (fun q : Joint (ι := ι) => futureArray s q.1) inferInstance) | |
| (joinedFiltration s) (rho ⊗ₘ (eta ×ₖ eta)) := by | |
| exact (ContinuousNoiseSplit.indepFun_future_conditionalCopies s rho eta | |
| (stoppedArray s) (measurable_stoppedArray s) hind).symm | |
| theorem indep_increment_joinedFiltration [Countable ι] | |
| (rho : Measure (ι → Path)) [IsProbabilityMeasure rho] | |
| (eta : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel eta] | |
| {s t : ℝ≥0} (hst : s ≤ t) | |
| (hind : IndepFun (fun q : (ι → Path) × (ι → Path) => | |
| (stoppedArray s q.1, stoppedArray s q.2)) | |
| (fun q => futureArray s q.1) (rho ⊗ₘ eta)) : | |
| Indep (MeasurableSpace.comap | |
| (fun q : Joint (ι := ι) => fun i => q.1 i t - q.1 i s) inferInstance) | |
| (joinedFiltration s) (rho ⊗ₘ (eta ×ₖ eta)) := by | |
| have hcopy := ContinuousNoiseSplit.indepFun_future_conditionalCopies s rho eta | |
| (stoppedArray s) (measurable_stoppedArray s) hind | |
| have heval : Measurable (fun w : ι → Path => fun i => w i (t - s)) := | |
| Measurable.of_eval fun i => | |
| (continuous_eval_const (t - s)).measurable.comp (measurable_pi_apply i) | |
| have hi := hcopy.symm.comp heval measurable_id | |
| change IndepFun (fun q : Joint (ι := ι) => fun i => q.1 i t - q.1 i s) | |
| (fun q => (stoppedArray s q.1, (stoppedArray s q.2.1, stoppedArray s q.2.2))) | |
| (rho ⊗ₘ (eta ×ₖ eta)) | |
| simpa only [Function.comp_def, id_eq, futureArray, futurePath_apply, | |
| add_tsub_cancel_of_le hst] using hi | |
| end Problem367.ContinuousCopyBrownian | |
| namespace Problem367.ConstructedContinuousCopies | |
| open WienerSampler ContinuousObservationHistory ContinuousNoiseSplit | |
| theorem exists_conditionalCopies_independentIncrements | |
| {ι Ω : Type*} [Countable ι] [mΩ : MeasurableSpace Ω] | |
| (P : Measure Ω) [IsProbabilityMeasure P] (F : Filtration ℝ≥0 mΩ) | |
| (W Y : Ω → ι → Path) | |
| (hW : ∀ i t, Measurable[F t] (fun ω => W ω i t)) | |
| (hY : ∀ i t, Measurable[F t] (fun ω => Y ω i t)) | |
| (hB : ∀ i, IsPreBrownianReal (fun t ω => W ω i t) P) | |
| (hind : ∀ s t, s ≤ t → Indep | |
| (MeasurableSpace.comap (fun ω i => W ω i t - W ω i s) inferInstance) (F s) P) : | |
| ∃ (rho : Measure (ι → Path)) (eta : Kernel (ι → Path) (ι → Path)), | |
| IsProbabilityMeasure rho ∧ IsMarkovKernel eta ∧ | |
| rho = P.map W ∧ rho ⊗ₘ eta = P.map (fun ω => (W ω, Y ω)) ∧ | |
| (∀ i, IsBrownianReal | |
| (fun t (q : ContinuousCopyBrownian.Joint (ι := ι)) => q.1 i t) | |
| (rho ⊗ₘ (eta ×ₖ eta))) ∧ | |
| (∀ i, Martingale | |
| (fun t (q : ContinuousCopyBrownian.Joint (ι := ι)) => q.1 i t) | |
| ContinuousCopyBrownian.joinedFiltration (rho ⊗ₘ (eta ×ₖ eta))) ∧ | |
| (∀ s t, s ≤ t → Indep | |
| (MeasurableSpace.comap | |
| (fun q : ContinuousCopyBrownian.Joint (ι := ι) => | |
| fun i => q.1 i t - q.1 i s) inferInstance) | |
| (ContinuousCopyBrownian.joinedFiltration s) (rho ⊗ₘ (eta ×ₖ eta))) ∧ | |
| MeasurePreserving (Prod.map id Prod.fst : ContinuousCopyBrownian.Joint (ι := ι) → _) | |
| (rho ⊗ₘ (eta ×ₖ eta)) (P.map (fun ω => (W ω, Y ω))) ∧ | |
| MeasurePreserving (Prod.map id Prod.snd : ContinuousCopyBrownian.Joint (ι := ι) → _) | |
| (rho ⊗ₘ (eta ×ₖ eta)) (P.map (fun ω => (W ω, Y ω))) := by | |
| obtain ⟨rho, eta, hrhop, hetap, hrho, hjoint, hBc, hMc⟩ := | |
| exists_compatible_conditionalCopies P F W Y hW hY hB hind | |
| let : IsProbabilityMeasure rho := hrhop | |
| let : IsMarkovKernel eta := hetap | |
| refine ⟨rho, eta, hrhop, hetap, hrho, hjoint, hBc, hMc, ?_, ?_, ?_⟩ | |
| · intro s t hst | |
| apply ContinuousCopyBrownian.indep_increment_joinedFiltration rho eta hst | |
| rw [hjoint] | |
| have hWm : Measurable W := Measurable.of_eval fun i => | |
| measurable_path_of_eval (fun u => (hW i u).mono (F.le u) le_rfl) | |
| have hYm : Measurable Y := Measurable.of_eval fun i => | |
| measurable_path_of_eval (fun u => (hY i u).mono (F.le u) le_rfl) | |
| apply ConstructedConditionalCopies.indepFun_map_of_comp (hWm.prodMk hYm) | |
| (((measurable_stoppedArray s).comp measurable_fst).prodMk | |
| ((measurable_stoppedArray s).comp measurable_snd)) | |
| ((measurable_futureArray s).comp measurable_fst) | |
| exact indepFun_continuous_future P F W Y hW hY hind s | |
| · rw [← hjoint] | |
| exact measurePreserving_conditionalCopy_fst rho eta | |
| · rw [← hjoint] | |
| exact measurePreserving_conditionalCopy_snd rho eta | |
| end Problem367.ConstructedContinuousCopies | |
| end | |
| end OAI | |