File size: 3,224 Bytes
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 | import OAI.Probability.FreeIsing.Martingales.ContinuousSubsequence
import OAI.Probability.FreeIsing.Martingales.PredictableVersion
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped NNReal Topology
namespace Problem367.PredictableMesh
def repair {Ω : Type*} (u : ℝ≥0 → Ω → ℝ) (t : ℝ≥0) (ω : Ω) : ℝ :=
limUnder atTop (fun n => sample u n t ω)
theorem repair_isStronglyPredictable {Ω : Type*} [mΩ : MeasurableSpace Ω]
{F : Filtration ℝ≥0 mΩ} {u : ℝ≥0 → Ω → ℝ}
(hu : StronglyAdapted F u) : IsStronglyPredictable F (repair u) := by
let : MeasurableSpace (ℝ≥0 × Ω) := F.predictable
exact StronglyMeasurable.limUnder (fun n => sample_isStronglyPredictable hu n)
theorem repair_eq_of_continuousOn {Ω : Type*} {u : ℝ≥0 → Ω → ℝ}
{T : ℝ≥0} {ω : Ω} (hc : ContinuousOn (fun t => u t ω) (Iic T))
{t : ℝ≥0} (ht : t ≤ T) : repair u t ω = u t ω := by
apply Filter.Tendsto.limUnder_eq
apply (hc t ht).tendsto.comp
apply tendsto_nhdsWithin_iff.mpr
exact ⟨tendsto_left t, Filter.Eventually.of_forall
(fun n => (left_le n t).trans ht)⟩
end Problem367.PredictableMesh
namespace Problem367.MartingaleSquareMaximal
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
{F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ}
theorem exists_predictable_continuous_subsequence_of_terminal_L2_cauchy
(hM : ∀ n, Martingale (M n) F μ)
(h2 : ∀ n t, MemLp (M n t) 2 μ)
(hc : ∀ n, ∀ᵐ ω ∂μ, Continuous (fun t => M n t ω)) (T : ℝ≥0)
(hC : CauchySeq (fun n => (h2 n T).toLp (M n T))) :
∃ (v : ℕ → ℕ) (L : ℝ≥0 → Ω → ℝ), StrictMono v ∧
IsStronglyPredictable F L ∧
(∀ᵐ ω ∂μ, ContinuousOn (fun t => L t ω) (Iic T)) ∧
(∀ᵐ ω ∂μ, ∀ t ≤ T,
Tendsto (fun n => M (v n) t ω) atTop (𝓝 (L t ω))) := by
obtain ⟨v, hv, hlim⟩ :=
exists_continuous_subsequence_of_terminal_L2_cauchy hM h2 hc T hC
let R : ℝ≥0 → Ω → ℝ := fun t ω => limUnder atTop (fun n => M (v n) t ω)
have hR : StronglyAdapted F R := fun t => by
let : MeasurableSpace Ω := F t
exact StronglyMeasurable.limUnder (fun n => (hM (v n)).stronglyMeasurable t)
have hpath : ∀ᵐ ω ∂μ, ContinuousOn (fun t => R t ω) (Iic T) ∧
∀ t ≤ T, Tendsto (fun n => M (v n) t ω) atTop (𝓝 (R t ω)) := by
filter_upwards [hlim] with ω hω
obtain ⟨f, hf⟩ := hω
have heq (t : Iic T) : R t ω = f t := (hf t).limUnder_eq
constructor
· rw [continuousOn_iff_continuous_domRestrict]
exact f.continuous.congr (fun t => (heq t).symm)
· intro t ht
simpa only [heq ⟨t, ht⟩] using hf ⟨t, ht⟩
refine ⟨v, PredictableMesh.repair R, hv,
PredictableMesh.repair_isStronglyPredictable hR, ?_, ?_⟩
· filter_upwards [hpath] with ω hω
exact hω.1.congr (fun t ht => PredictableMesh.repair_eq_of_continuousOn hω.1 ht)
· filter_upwards [hpath] with ω hω
intro t ht
rw [PredictableMesh.repair_eq_of_continuousOn hω.1 ht]
exact hω.2 t ht
end Problem367.MartingaleSquareMaximal
end
end OAI
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