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b317718 | 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.CopyFiltration
import OAI.Probability.FreeIsing.Stochastic.WienerSampler
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.ContinuousObservationHistory
open WienerSampler
def stoppedPath (s : ℝ≥0) (f : Path) : Path :=
⟨fun t => f (min t s), f.continuous.comp (continuous_id.min continuous_const)⟩
@[simp]
theorem stoppedPath_apply (s : ℝ≥0) (f : Path) (t : ℝ≥0) :
stoppedPath s f t = f (min t s) := rfl
theorem measurable_stoppedPath (s : ℝ≥0) : Measurable (stoppedPath s) := by
apply measurable_path_of_eval
intro t
exact (continuous_eval_const (min t s)).measurable
theorem stoppedPath_stoppedPath {s t : ℝ≥0} (hst : s ≤ t) (f : Path) :
stoppedPath s (stoppedPath t f) = stoppedPath s f := by
ext u
simp only [stoppedPath_apply, min_eq_left ((min_le_right u s).trans hst)]
variable {ι : Type*}
def stoppedArray (s : ℝ≥0) (f : ι → Path) : ι → Path :=
fun i => stoppedPath s (f i)
theorem measurable_stoppedArray (s : ℝ≥0) : Measurable (stoppedArray (ι := ι) s) := by
exact Measurable.of_eval fun index =>
(measurable_stoppedPath s).comp (measurable_pi_apply index)
theorem stoppedArray_stoppedArray {s t : ℝ≥0} (hst : s ≤ t) (f : ι → Path) :
stoppedArray s (stoppedArray t f) = stoppedArray s f := by
funext i
exact stoppedPath_stoppedPath hst (f i)
theorem monotone_stoppedArray_comap : Monotone
(fun s => MeasurableSpace.comap (stoppedArray (ι := ι) s) inferInstance) := by
intro s t hst
have h := MeasurableSpace.comap_mono (g := stoppedArray (ι := ι) t)
(measurable_stoppedArray (ι := ι) s).comap_le
simpa only [MeasurableSpace.comap_comp, Function.comp_def,
stoppedArray_stoppedArray hst] using h
theorem measurable_stoppedArray_of_eval {Ω : Type*} [mΩ : MeasurableSpace Ω]
(s : ℝ≥0) (F : Ω → ι → Path)
(hF : ∀ i t, t ≤ s → Measurable (fun ω => F ω i t)) :
Measurable (fun ω => stoppedArray s (F ω)) := by
apply Measurable.of_eval
intro i
apply measurable_path_of_eval
intro t
exact hF i (min t s) (min_le_right t s)
theorem measurable_past_of_stoppedArray {Ω : Type*} [mΩ : MeasurableSpace Ω]
(s : ℝ≥0) (F : Ω → ι → Path)
(hF : Measurable (fun ω => stoppedArray s (F ω))) :
Measurable (fun ω i (t : Set.Iic s) => F ω i t) := by
apply Measurable.of_eval
intro i
apply Measurable.of_eval
intro t
have h := (continuous_eval_const (t : ℝ≥0)).measurable.comp
((measurable_pi_apply i).comp hF)
simpa only [Function.comp_def, stoppedArray, stoppedPath_apply,
min_eq_left (show (t : ℝ≥0) ≤ s from t.property)] using h
theorem stoppedArray_comap_eq_past (s : ℝ≥0) :
MeasurableSpace.comap (stoppedArray (ι := ι) s) inferInstance =
MeasurableSpace.comap
(fun f : ι → Path => fun i (t : Set.Iic s) => f i t) inferInstance := by
apply le_antisymm
· let m := MeasurableSpace.comap
(fun f : ι → Path => fun i (t : Set.Iic s) => f i t) inferInstance
have hp : @Measurable (ι → Path) (ι → Set.Iic s → ℝ) m inferInstance
(fun f i t => f i t) := comap_measurable _
have hs := measurable_stoppedArray_of_eval (mΩ := m) s (fun f : ι → Path => f)
(fun i t ht => (measurable_pi_apply (⟨t, ht⟩ : Set.Iic s)).comp
((measurable_pi_apply i).comp hp))
exact hs.comap_le
· exact (measurable_past_of_stoppedArray
(mΩ := MeasurableSpace.comap (stoppedArray (ι := ι) s) inferInstance)
s (fun f : ι → Path => f) (comap_measurable _)).comap_le
theorem stoppedArray_comap_eq_natural (s : ℝ≥0) :
MeasurableSpace.comap (stoppedArray (ι := ι) s) inferInstance =
⨆ i : ι, Filtration.natural (fun t (f : ι → Path) => f i t)
(fun t => ((continuous_eval_const t).measurable.comp
(measurable_pi_apply i)).stronglyMeasurable) s := by
rw [stoppedArray_comap_eq_past]
simp only [MeasurableSpace.comap_process_pi, Filtration.natural_eq_comap]
def joinedFiltration : Filtration ℝ≥0
(inferInstance : MeasurableSpace ((ℝ≥0 → ι → ℝ) × ((ι → Path) × (ι → Path)))) :=
ConditionalCopyBrownian.joinedFiltration stoppedArray measurable_stoppedArray
monotone_stoppedArray_comap
theorem commonNoise_martingale [Countable ι]
(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) (ι → Path)) [IsMarkovKernel eta]
(hind : ∀ s, IndepFun
(fun q : (ℝ≥0 → ι → ℝ) × (ι → Path) =>
(PathSplit.pastPath s q.1, stoppedArray s q.2))
(fun q => PathSplit.futureIncrements s q.1) (rho ⊗ₘ eta)) :
Martingale (fun t (q : (ℝ≥0 → ι → ℝ) × ((ι → Path) × (ι → Path))) => q.1 t i)
joinedFiltration (rho ⊗ₘ (eta ×ₖ eta)) :=
ConditionalCopyArrayBrownian.commonNoise_martingale_joinedFiltration i rho hB eta
stoppedArray measurable_stoppedArray monotone_stoppedArray_comap hind
end Problem367.ContinuousObservationHistory
end
end OAI
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