| import OAI.Probability.FreeIsing.Conditioning.CopyImmersion |
| import OAI.Probability.FreeIsing.Martingales.Pushforward |
| import OAI.Probability.FreeIsing.Stochastic.ContinuousCopyBrownian |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal |
|
|
| namespace Problem367.ContinuousCopyBrownian |
| open WienerSampler ContinuousObservationHistory ContinuousNoiseSplit |
|
|
| variable {ι : Type*} |
|
|
| def singleHistory (s : ℝ≥0) (q : (ι → Path) × (ι → Path)) := |
| (stoppedArray s q.1, stoppedArray s q.2) |
|
|
| lemma measurable_singleHistory (s : ℝ≥0) : Measurable (singleHistory (ι := ι) s) := |
| ((measurable_stoppedArray s).comp measurable_fst).prodMk |
| ((measurable_stoppedArray s).comp measurable_snd) |
|
|
| lemma singleHistory_singleHistory {s t : ℝ≥0} (hst : s ≤ t) |
| (q : (ι → Path) × (ι → Path)) : singleHistory s (singleHistory t q) = singleHistory s q := by |
| simp only [singleHistory, stoppedArray_stoppedArray hst] |
|
|
| def singleFiltration : Filtration ℝ≥0 |
| (inferInstance : MeasurableSpace ((ι → Path) × (ι → Path))) where |
| seq s := MeasurableSpace.comap (singleHistory s) inferInstance |
| mono' := by |
| intro s t hst |
| have h := MeasurableSpace.comap_mono (g := singleHistory (ι := ι) t) |
| (measurable_singleHistory s).comap_le |
| simpa only [MeasurableSpace.comap_comp, Function.comp_def, |
| singleHistory_singleHistory hst] using h |
| le' s := (measurable_singleHistory s).comap_le |
|
|
| lemma measurable_noisePast_singleFiltration (s : ℝ≥0) : |
| Measurable[singleFiltration (ι := ι) s] |
| (fun q : (ι → Path) × (ι → Path) => stoppedArray s q.1) := |
| measurable_fst.comp (comap_measurable (singleHistory s)) |
|
|
| lemma joinedFiltration_eq_left (s : ℝ≥0) : |
| joinedFiltration (ι := ι) s = |
| (singleFiltration s).comap (fun q : Joint (ι := ι) => (q.1, q.2.1)) ⊔ |
| MeasurableSpace.comap (fun q : Joint (ι := ι) => stoppedArray s q.2.2) inferInstance := by |
| change MeasurableSpace.comap |
| (fun q : Joint (ι := ι) => |
| (stoppedArray s q.1, (stoppedArray s q.2.1, stoppedArray s q.2.2))) |
| ((inferInstance : MeasurableSpace (ι → Path)).prod |
| ((inferInstance : MeasurableSpace (ι → Path)).prod |
| (inferInstance : MeasurableSpace (ι → Path)))) = _ |
| simp only [singleFiltration, singleHistory, MeasurableSpace.comap_comp, |
| Function.comp_def] |
| change MeasurableSpace.comap |
| (fun q : Joint (ι := ι) => |
| (stoppedArray s q.1, (stoppedArray s q.2.1, stoppedArray s q.2.2))) |
| ((inferInstance : MeasurableSpace (ι → Path)).prod |
| ((inferInstance : MeasurableSpace (ι → Path)).prod |
| (inferInstance : MeasurableSpace (ι → Path)))) = |
| MeasurableSpace.comap |
| (fun q : Joint (ι := ι) => (stoppedArray s q.1, stoppedArray s q.2.1)) |
| ((inferInstance : MeasurableSpace (ι → Path)).prod |
| (inferInstance : MeasurableSpace (ι → Path))) ⊔ |
| MeasurableSpace.comap (fun q : Joint (ι := ι) => stoppedArray s q.2.2) inferInstance |
| rw [MeasurableSpace.comap_prodMk, MeasurableSpace.comap_prodMk, |
| MeasurableSpace.comap_prodMk, sup_assoc] |
|
|
| theorem exists_past_kernel [Countable ι] |
| (ρ : Measure (ι → Path)) [IsProbabilityMeasure ρ] |
| (η : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel η] (s : ℝ≥0) |
| (hind : IndepFun (singleHistory s) |
| (fun q : (ι → Path) × (ι → Path) => futureArray s q.1) (ρ ⊗ₘ η)) : |
| ∃ κ : Kernel (ι → Path) (ι → Path), IsMarkovKernel κ ∧ |
| (fun w => (η.map (stoppedArray s)) w) =ᵐ[ρ] |
| fun w => κ (stoppedArray s w) := |
| ConditionalCopyImmersion.exists_past_kernel_of_reconstructible_split ρ |
| (stoppedArray s) (measurable_stoppedArray s) |
| (futureArray s) (measurable_futureArray s) (measurableEmbedding_split s) |
| η (stoppedArray s) (measurable_stoppedArray s) hind |
|
|
| theorem martingale_left_of_future_indep [Countable ι] |
| (ρ : Measure (ι → Path)) [IsProbabilityMeasure ρ] |
| (η : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel η] |
| (hind : ∀ s, IndepFun (singleHistory s) |
| (fun q : (ι → Path) × (ι → Path) => futureArray s q.1) (ρ ⊗ₘ η)) |
| {M : ℝ≥0 → (ι → Path) × (ι → Path) → ℝ} |
| (hM : Martingale M singleFiltration (ρ ⊗ₘ η)) : |
| Martingale (fun t (q : Joint (ι := ι)) => M t (q.1, q.2.1)) |
| joinedFiltration (ρ ⊗ₘ (η ×ₖ η)) := by |
| exact ConditionalCopyImmersion.martingale_left ρ η |
| stoppedArray measurable_stoppedArray stoppedArray measurable_stoppedArray |
| singleFiltration joinedFiltration joinedFiltration_eq_left |
| measurable_noisePast_singleFiltration |
| (fun s => exists_past_kernel ρ η s (hind s)) hM |
|
|
| lemma measurable_swap_joinedFiltration (s : ℝ≥0) : |
| Measurable[joinedFiltration (ι := ι) s, joinedFiltration s] |
| (fun q : Joint (ι := ι) => (q.1, (q.2.2, q.2.1))) := by |
| apply Measurable.of_comap_le |
| change (MeasurableSpace.comap (history s) |
| (inferInstance : MeasurableSpace (Joint (ι := ι)))).comap |
| (fun q : Joint (ι := ι) => (q.1, (q.2.2, q.2.1))) ≤ _ |
| rw [MeasurableSpace.comap_comp] |
| have hh : Measurable[joinedFiltration (ι := ι) s] (history s) := comap_measurable _ |
| have hm := (measurable_fst.comp hh).prodMk |
| (((measurable_snd.comp measurable_snd).comp hh).prodMk |
| ((measurable_fst.comp measurable_snd).comp hh)) |
| exact hm.comap_le |
|
|
| theorem martingale_right_of_future_indep [Countable ι] |
| (ρ : Measure (ι → Path)) [IsProbabilityMeasure ρ] |
| (η : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel η] |
| (hind : ∀ s, IndepFun (singleHistory s) |
| (fun q : (ι → Path) × (ι → Path) => futureArray s q.1) (ρ ⊗ₘ η)) |
| {M : ℝ≥0 → (ι → Path) × (ι → Path) → ℝ} |
| (hM : Martingale M singleFiltration (ρ ⊗ₘ η)) : |
| Martingale (fun t (q : Joint (ι := ι)) => M t (q.1, q.2.2)) |
| joinedFiltration (ρ ⊗ₘ (η ×ₖ η)) := by |
| have hl := martingale_left_of_future_indep ρ η hind hM |
| apply MartingalePushforward.martingale_of_pullback |
| (ConditionalCopyImmersion.swap_measurePreserving ρ η) |
| (F := joinedFiltration) (G := joinedFiltration) |
| (fun t => (measurable_swap_joinedFiltration t).comap_le) |
| (fun t => (hl.stronglyAdapted t).comp_measurable (measurable_swap_joinedFiltration t)) |
| exact hl |
|
|
| end Problem367.ContinuousCopyBrownian |
|
|
| end |
|
|
| end OAI |
|
|
