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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 | import OAI.Probability.FreeIsing.Conditioning.CopyArrayBrownian
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal
namespace Problem367.ConditionalCopyBrownian
variable {E O X : Type*} [MeasurableSpace E] [MeasurableSpace X]
theorem pastPath_comap_mono {s t : ℝ≥0} (hst : s ≤ t) :
MeasurableSpace.comap (PathSplit.pastPath (E := E) s) inferInstance ≤
MeasurableSpace.comap (PathSplit.pastPath (E := E) t) inferInstance := by
let : MeasurableSpace (ℝ≥0 → E) :=
MeasurableSpace.comap (PathSplit.pastPath (E := E) t)
(inferInstance : MeasurableSpace (Set.Iic t → E))
have hm : Measurable (PathSplit.pastPath (E := E) s) := by
exact measurable_pi_iff.mpr fun u =>
(measurable_pi_apply (⟨u.1, u.2.trans hst⟩ : Set.Iic t)).comp
(comap_measurable (PathSplit.pastPath t))
exact hm.comap_le
theorem joinedPast_eq_sup (s : ℝ≥0) (p : O → X) :
joinedPast (E := E) s p =
MeasurableSpace.comap (fun z : (ℝ≥0 → E) × (O × O) => PathSplit.pastPath s z.1) inferInstance ⊔
(MeasurableSpace.comap (fun z : (ℝ≥0 → E) × (O × O) => p z.2.1) inferInstance ⊔
MeasurableSpace.comap (fun z : (ℝ≥0 → E) × (O × O) => p z.2.2) inferInstance) := by
have h : joinedPast (E := E) s p =
MeasurableSpace.comap (fun z : (ℝ≥0 → E) × (O × O) => PathSplit.pastPath s z.1) inferInstance ⊔
MeasurableSpace.comap (fun z : (ℝ≥0 → E) × (O × O) => (p z.2.1, p z.2.2)) inferInstance :=
MeasurableSpace.comap_prodMk _ _
rw [h]
congr 1
exact MeasurableSpace.comap_prodMk _ _
theorem joinedPast_mono {s t : ℝ≥0} (hst : s ≤ t) {p q : O → X}
(hpq : MeasurableSpace.comap p inferInstance ≤
MeasurableSpace.comap q inferInstance) :
joinedPast (E := E) s p ≤ joinedPast t q := by
rw [joinedPast_eq_sup, joinedPast_eq_sup]
apply sup_le_sup
· have h := MeasurableSpace.comap_mono (g := fun z : (ℝ≥0 → E) × (O × O) => z.1)
(pastPath_comap_mono (E := E) hst)
simpa only [MeasurableSpace.comap_comp, Function.comp_def] using h
· apply sup_le_sup
· have h := MeasurableSpace.comap_mono (g := fun z : (ℝ≥0 → E) × (O × O) => z.2.1) hpq
simpa only [MeasurableSpace.comap_comp, Function.comp_def] using h
· have h := MeasurableSpace.comap_mono (g := fun z : (ℝ≥0 → E) × (O × O) => z.2.2) hpq
simpa only [MeasurableSpace.comap_comp, Function.comp_def] using h
variable [MeasurableSpace O]
def joinedFiltration (past : ℝ≥0 → O → X) (hpast : ∀ s, Measurable (past s))
(hmono : Monotone (fun s => MeasurableSpace.comap (past s) inferInstance)) :
Filtration ℝ≥0 (inferInstance : MeasurableSpace ((ℝ≥0 → E) × (O × O))) where
seq s := joinedPast s (past s)
mono' := fun _ _ hst => joinedPast_mono hst (hmono hst)
le' s := joinedPast_le s (hpast s)
@[simp]
theorem joinedFiltration_apply (past : ℝ≥0 → O → X) (hpast : ∀ s, Measurable (past s))
(hmono : Monotone (fun s => MeasurableSpace.comap (past s) inferInstance)) (s : ℝ≥0) :
joinedFiltration (E := E) past hpast hmono s = joinedPast s (past s) := rfl
end Problem367.ConditionalCopyBrownian
namespace Problem367.ConditionalCopyArrayBrownian
open ConditionalCopyBrownian
variable {ι O X : Type*} [MeasurableSpace O] [MeasurableSpace X]
variable [StandardBorelSpace X] [Nonempty X]
theorem commonNoise_martingale_joinedFiltration
(i : ι) (rho : Measure (ℝ≥0 → ι → ℝ)) [IsProbabilityMeasure rho]
(hB : IsPreBrownianReal (fun t (w : ℝ≥0 → ι → ℝ) => w t i) rho)
(eta : Kernel (ℝ≥0 → ι → ℝ) O) [IsMarkovKernel eta]
(past : ℝ≥0 → O → X) (hpast : ∀ s, Measurable (past s))
(hmono : Monotone (fun s => MeasurableSpace.comap (past s) inferInstance))
(hind : ∀ s, IndepFun
(fun q : (ℝ≥0 → ι → ℝ) × O => (PathSplit.pastPath s q.1, past s q.2))
(fun q => PathSplit.futureIncrements s q.1) (rho ⊗ₘ eta)) :
Martingale (fun t (q : (ℝ≥0 → ι → ℝ) × (O × O)) => q.1 t i)
(joinedFiltration past hpast hmono) (rho ⊗ₘ (eta ×ₖ eta)) :=
commonNoise_martingale i rho hB eta past hpast
(joinedFiltration past hpast hmono) (fun _ => rfl) hind
theorem isBrownian_commonNoise
(i : ι) (rho : Measure (ℝ≥0 → ι → ℝ)) [IsProbabilityMeasure rho]
(hB : IsBrownianReal (fun t (w : ℝ≥0 → ι → ℝ) => w t i) rho)
(eta : Kernel (ℝ≥0 → ι → ℝ) O) [IsMarkovKernel eta] :
IsBrownianReal (fun t (q : (ℝ≥0 → ι → ℝ) × (O × O)) => q.1 t i)
(rho ⊗ₘ (eta ×ₖ eta)) := by
have hmp : MeasurePreserving (Prod.fst : (ℝ≥0 → ι → ℝ) × (O × O) → (ℝ≥0 → ι → ℝ))
(rho ⊗ₘ (eta ×ₖ eta)) rho := ⟨measurable_fst, Measure.fst_compProd _ _⟩
exact ⟨isPreBrownian_commonNoise i rho hB.toIsPreBrownianReal eta,
hmp.quasiMeasurePreserving.ae hB.cont⟩
end Problem367.ConditionalCopyArrayBrownian
end
end OAI
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