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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | 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
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