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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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
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