| import OAI.Probability.FreeIsing.Martingales.ContinuousL2Limit | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped NNReal Topology | |
| namespace Problem367.MartingaleSquareMaximal | |
| variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Preorder ι] | |
| {μ : Measure Ω} [IsFiniteMeasure μ] {F : Filtration ι mΩ} | |
| theorem condExp_ae_eq_of_L2_limits {M : ℕ → ι → Ω → ℝ} | |
| (hM : ∀ n, Martingale (M n) F μ) | |
| (h2 : ∀ n t, MemLp (M n t) 2 μ) | |
| {s t : ι} (hst : s ≤ t) {X Y : Ω → ℝ} | |
| (hX : MemLp X 2 μ) (hY : MemLp Y 2 μ) | |
| (hs : Tendsto (fun n => (h2 n s).toLp (M n s)) atTop (𝓝 (hX.toLp X))) | |
| (ht : Tendsto (fun n => (h2 n t).toLp (M n t)) atTop (𝓝 (hY.toLp Y))) : | |
| μ[Y | F s] =ᵐ[μ] X := by | |
| let C := condExpL2 ℝ ℝ (F.le s) (μ := μ) | |
| have hC : Continuous (fun f : Lp ℝ 2 μ => (C f : Lp ℝ 2 μ)) := | |
| continuous_subtype_val.comp C.continuous | |
| have heq (n : ℕ) : (C ((h2 n t).toLp (M n t)) : Lp ℝ 2 μ) = | |
| (h2 n s).toLp (M n s) := by | |
| apply Lp.ext | |
| exact ((h2 n t).condExpL2_ae_eq_condExp (𝕜 := ℝ) (F.le s)).trans | |
| (((hM n).condExp_ae_eq hst).trans (h2 n s).coeFn_toLp.symm) | |
| have hclass : (C (hY.toLp Y) : Lp ℝ 2 μ) = hX.toLp X := by | |
| apply tendsto_nhds_unique ((hC.tendsto _).comp ht) | |
| convert hs using 1 | |
| funext n | |
| exact heq n | |
| have hae : (C (hY.toLp Y) : Ω → ℝ) =ᵐ[μ] X := by | |
| change ((C (hY.toLp Y) : Lp ℝ 2 μ) : Ω → ℝ) =ᵐ[μ] X | |
| rw [hclass] | |
| exact hX.coeFn_toLp | |
| exact (hY.condExpL2_ae_eq_condExp (𝕜 := ℝ) (F.le s)).symm.trans hae | |
| theorem martingale_of_L2_limits {M : ℕ → ι → Ω → ℝ} {L : ι → Ω → ℝ} | |
| (hM : ∀ n, Martingale (M n) F μ) (h2 : ∀ n t, MemLp (M n t) 2 μ) | |
| (hL : StronglyAdapted F L) (hL2 : ∀ t, MemLp (L t) 2 μ) | |
| (hlim : ∀ t, Tendsto (fun n => (h2 n t).toLp (M n t)) atTop | |
| (𝓝 ((hL2 t).toLp (L t)))) : Martingale L F μ := by | |
| refine ⟨hL, fun s t hst => ?_⟩ | |
| exact condExp_ae_eq_of_L2_limits hM h2 hst (hL2 s) (hL2 t) (hlim s) (hlim t) | |
| theorem exists_predictable_continuous_L2_limit_condExp | |
| {F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ} | |
| (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))) : | |
| ∃ L : ℝ≥0 → Ω → ℝ, IsStronglyPredictable F L ∧ | |
| (∀ᵐ ω ∂μ, ContinuousOn (fun t => L t ω) (Iic T)) ∧ | |
| (∀ t ≤ T, ∃ hL2 : MemLp (L t) 2 μ, | |
| Tendsto (fun n => (h2 n t).toLp (M n t)) atTop (𝓝 (hL2.toLp (L t)))) ∧ | |
| ∀ s t, s ≤ t → t ≤ T → μ[L t | F s] =ᵐ[μ] L s := by | |
| obtain ⟨L, hpred, hc, hlim⟩ := exists_predictable_continuous_L2_limit hM h2 hc T hC | |
| refine ⟨L, hpred, hc, hlim, fun s t hst ht => ?_⟩ | |
| obtain ⟨hLs, hs⟩ := hlim s (hst.trans ht) | |
| obtain ⟨hLt, ht'⟩ := hlim t ht | |
| exact condExp_ae_eq_of_L2_limits hM h2 hst hLs hLt hs ht' | |
| end Problem367.MartingaleSquareMaximal | |
| end | |
| end OAI | |