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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Probability.FreeIsing.Conditioning.ConstructedConditionalCopies
import OAI.Probability.FreeIsing.Stochastic.ContinuousCopyBrownian
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.ConstructedContinuousCopies
open WienerSampler ContinuousObservationHistory ContinuousNoiseSplit
variable {ι Ω : Type*} [mΩ : MeasurableSpace Ω]
theorem measurable_futureArray_of_eval (s : ℝ≥0) (W : Ω → ι → Path)
(hW : ∀ i t, Measurable (fun ω => W ω i (s + t) - W ω i s)) :
Measurable (fun ω => futureArray s (W ω)) := by
apply Measurable.of_eval
intro i
apply measurable_path_of_eval
intro t
exact hW i t
theorem indepFun_continuous_future
(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))
(hind : ∀ s t, s ≤ t → Indep
(MeasurableSpace.comap (fun ω i => W ω i t - W ω i s) inferInstance) (F s) P)
(s : ℝ≥0) :
IndepFun (fun ω => (stoppedArray s (W ω), stoppedArray s (Y ω)))
(fun ω => futureArray s (W ω)) P := by
let X : ℝ≥0 → Ω → ι → ℝ := fun t ω i => W ω i t
have hX (t : ℝ≥0) : Measurable[F t] (X t) := by
let : MeasurableSpace Ω := F t
exact Measurable.of_eval fun i => hW i t
let mFuture := MeasurableSpace.comap
(fun ω (t : Set.Ici s) => X t ω - X s ω) inferInstance
have hRaw : @Measurable Ω (Set.Ici s → ι → ℝ) mFuture inferInstance
(fun ω t => X t ω - X s ω) := comap_measurable _
have hFut : @Measurable Ω (ι → Path) mFuture inferInstance
(fun ω => futureArray s (W ω)) :=
measurable_futureArray_of_eval (mΩ := mFuture) s W (fun i t =>
(measurable_pi_apply i).comp
((measurable_pi_apply (⟨s + t, by change s ≤ s + t; exact le_self_add⟩ : Set.Ici s)).comp hRaw))
have hPast : Measurable[F s]
(fun ω => (stoppedArray s (W ω), stoppedArray s (Y ω))) :=
(measurable_stoppedArray_of_eval (mΩ := F s) s W
(fun i t ht => (hW i t).mono (F.mono ht) le_rfl)).prodMk
(measurable_stoppedArray_of_eval (mΩ := F s) s Y
(fun i t ht => (hY i t).mono (F.mono ht) le_rfl))
have hi : Indep (F s) mFuture P :=
FilteredFutureIndependence.indep_future (mΩ := mΩ) (P := P) hX hind s
exact indep_of_indep_of_le (μ := P) hi hPast.comap_le hFut.comap_le
theorem exists_compatible_conditionalCopies [Countable ι]
(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))) := by
have hWm : Measurable W := Measurable.of_eval fun i =>
measurable_path_of_eval (fun t => (hW i t).mono (F.le t) le_rfl)
have hYm : Measurable Y := Measurable.of_eval fun i =>
measurable_path_of_eval (fun t => (hY i t).mono (F.le t) le_rfl)
let Z : Ω → (ι → Path) × (ι → Path) := fun ω => (W ω, Y ω)
have hZ : Measurable Z := hWm.prodMk hYm
let R := P.map Z
have : IsProbabilityMeasure R := inferInstance
let rho := R.fst
let eta := R.condKernel
have hrho : rho = P.map W := Measure.fst_map_prodMk hWm hYm
have hdecomp : rho ⊗ₘ eta = R := Measure.disintegrate R eta
have hBrho (i : ι) : IsPreBrownianReal (fun t (w : ι → Path) => w i t) rho := by
rw [hrho]
constructor
intro I
have hm : Measurable (fun w : ι → Path => I.restrict (fun t => w i t)) :=
Measurable.of_eval fun t =>
(continuous_eval_const t.1).measurable.comp (measurable_pi_apply i)
refine ⟨hm.aemeasurable, ?_⟩
rw [Measure.map_map hm hWm]
exact (hB i).hasLaw I |>.map_eq
have hOriginal (s : ℝ≥0) : IndepFun
(fun q : (ι → Path) × (ι → Path) => (stoppedArray s q.1, stoppedArray s q.2))
(fun q => futureArray s q.1) (rho ⊗ₘ eta) := by
rw [hdecomp]
apply ConstructedConditionalCopies.indepFun_map_of_comp hZ
(((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
exact ⟨rho, eta, inferInstance, inferInstance, hrho, hdecomp,
(fun i => ContinuousCopyBrownian.isBrownian_commonNoise i rho (hBrho i) eta),
(fun i => ContinuousCopyBrownian.commonNoise_martingale i rho (hBrho i) eta hOriginal)⟩
end Problem367.ConstructedContinuousCopies
end
end OAI
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