| import OAI.Probability.FreeIsing.Gaussian.ContinuousNoiseSplit |
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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.ContinuousCopyBrownian |
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| open WienerSampler ContinuousObservationHistory ContinuousNoiseSplit |
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| variable {ι : Type*} |
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| abbrev Joint := (ι → Path) × ((ι → Path) × (ι → Path)) |
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| def history (s : ℝ≥0) (q : Joint (ι := ι)) : Joint (ι := ι) := |
| (stoppedArray s q.1, (stoppedArray s q.2.1, stoppedArray s q.2.2)) |
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| theorem measurable_history (s : ℝ≥0) : Measurable (history (ι := ι) s) := |
| ((measurable_stoppedArray s).comp measurable_fst).prodMk |
| (((measurable_stoppedArray s).comp (measurable_fst.comp measurable_snd)).prodMk |
| ((measurable_stoppedArray s).comp (measurable_snd.comp measurable_snd))) |
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| theorem history_history {s t : ℝ≥0} (hst : s ≤ t) (q : Joint (ι := ι)) : |
| history s (history t q) = history s q := by |
| simp only [history, stoppedArray_stoppedArray hst] |
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| abbrev pastSpace (s : ℝ≥0) : MeasurableSpace (Joint (ι := ι)) := |
| MeasurableSpace.comap (history s) inferInstance |
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| def joinedFiltration : Filtration ℝ≥0 (inferInstance : MeasurableSpace (Joint (ι := ι))) where |
| seq := pastSpace |
| mono' := by |
| intro s t hst |
| have h := MeasurableSpace.comap_mono (g := history (ι := ι) t) (measurable_history s).comap_le |
| simpa only [MeasurableSpace.comap_comp, Function.comp_def, history_history hst] using h |
| le' s := (measurable_history s).comap_le |
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| theorem isBrownian_commonNoise |
| (i : ι) (rho : Measure (ι → Path)) [IsProbabilityMeasure rho] |
| (hB : IsPreBrownianReal (fun t (w : ι → Path) => w i t) rho) |
| (eta : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel eta] : |
| IsBrownianReal (fun t (q : Joint (ι := ι)) => q.1 i t) (rho ⊗ₘ (eta ×ₖ eta)) := by |
| have hmp : MeasurePreserving (Prod.fst : Joint (ι := ι) → (ι → Path)) |
| (rho ⊗ₘ (eta ×ₖ eta)) rho := ⟨measurable_fst, Measure.fst_compProd _ _⟩ |
| exact ⟨⟨fun I => (hB.hasLaw I).comp hmp.hasLaw⟩, |
| Filter.Eventually.of_forall (fun q => (q.1 i).continuous)⟩ |
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| theorem condExp_increment_eq_zero [Countable ι] |
| (i : ι) (rho : Measure (ι → Path)) [IsProbabilityMeasure rho] |
| (hB : IsPreBrownianReal (fun t (w : ι → Path) => w i t) 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)) : |
| (rho ⊗ₘ (eta ×ₖ eta))[(fun q : Joint (ι := ι) => q.1 i t - q.1 i s) | |
| pastSpace s] =ᵐ[rho ⊗ₘ (eta ×ₖ eta)] fun _ => 0 := by |
| have hcopy := ContinuousNoiseSplit.indepFun_future_conditionalCopies s rho eta |
| (stoppedArray s) (measurable_stoppedArray s) hind |
| have hf : Measurable (fun w : ι → Path => w i (t - s)) := |
| (continuous_eval_const (t - s)).measurable.comp (measurable_pi_apply i) |
| have hi : IndepFun (fun q : Joint (ι := ι) => q.1 i t - q.1 i s) (history s) |
| (rho ⊗ₘ (eta ×ₖ eta)) := by |
| change IndepFun (fun q : Joint (ι := ι) => q.1 i t - q.1 i s) |
| (fun q => (stoppedArray s q.1, (stoppedArray s q.2.1, stoppedArray s q.2.2))) _ |
| simpa only [Function.comp_def, id_eq, futureArray, futurePath_apply, |
| add_tsub_cancel_of_le hst] using hcopy.symm.comp hf measurable_id |
| have hm : Measurable (fun q : Joint (ι := ι) => q.1 i t - q.1 i s) := |
| (((continuous_eval_const t).measurable.comp (measurable_pi_apply i)).comp measurable_fst).sub |
| (((continuous_eval_const s).measurable.comp (measurable_pi_apply i)).comp measurable_fst) |
| have hc := condExp_indep_eq hm.comap_le (measurable_history s).comap_le |
| (measurable_iff_comap_le.mpr le_rfl).stronglyMeasurable hi |
| have hBc := (isBrownian_commonNoise i rho hB eta).toIsPreBrownianReal |
| have hz : (∫ q : Joint (ι := ι), q.1 i t - q.1 i s ∂rho ⊗ₘ (eta ×ₖ eta)) = 0 := by |
| rw [integral_sub (hBc.integrable_eval t) (hBc.integrable_eval s), |
| hBc.integral_eval, hBc.integral_eval, sub_self] |
| simpa only [hz] using hc |
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| theorem commonNoise_martingale [Countable ι] |
| (i : ι) (rho : Measure (ι → Path)) [IsProbabilityMeasure rho] |
| (hB : IsPreBrownianReal (fun t (w : ι → Path) => w i t) rho) |
| (eta : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel eta] |
| (hind : ∀ s, IndepFun (fun q : (ι → Path) × (ι → Path) => |
| (stoppedArray s q.1, stoppedArray s q.2)) |
| (fun q => futureArray s q.1) (rho ⊗ₘ eta)) : |
| Martingale (fun t (q : Joint (ι := ι)) => q.1 i t) joinedFiltration |
| (rho ⊗ₘ (eta ×ₖ eta)) := by |
| have had : StronglyAdapted (joinedFiltration (ι := ι)) |
| (fun t (q : Joint (ι := ι)) => q.1 i t) := by |
| intro t |
| change StronglyMeasurable[pastSpace (ι := ι) t] (fun q : Joint (ι := ι) => q.1 i t) |
| have hh : Measurable[pastSpace (ι := ι) t] (history (ι := ι) t) := comap_measurable _ |
| 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.stronglyMeasurable |
| apply martingale_of_conditional_increment_zero had |
| (fun t => (isBrownian_commonNoise i rho hB eta).toIsPreBrownianReal.integrable_eval t) |
| intro s t hst |
| exact condExp_increment_eq_zero i rho hB eta hst (hind s) |
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| end Problem367.ContinuousCopyBrownian |
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| end |
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| end OAI |
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