| import OAI.Probability.FreeIsing.Brownian.RowComparisonConstruction |
| import OAI.Probability.FreeIsing.Brownian.SpatialPrimitive |
| import OAI.Probability.FreeIsing.Stochastic.StoppedIntegralAssociativity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
|
|
| namespace Problem367.BrownianArrayFiltration |
|
|
| variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
|
|
| theorem compensatedSquare_of_spatial_integral_limits |
| [Countable ι] [IsProbabilityMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {B H I : ι → ℝ≥0 → Ω → ℝ} |
| {Y : ℝ≥0 → Ω → ℝ} |
| (hB : ∀ a, IsPreBrownianReal (B a) P) |
| (hAd : ∀ a, StronglyAdapted F (B a)) |
| (hpast : ∀ s t, s ≤ t → Indep (MeasurableSpace.comap |
| (fun ω a => B a t ω - B a s ω) inferInstance) (F s) P) |
| (hind : ∀ s t, s ≤ t → Pairwise fun a b => |
| IndepFun (fun ω => B a t ω - B a s ω) |
| (fun ω => B b t ω - B b s ω) P) |
| (hBc : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => B a t ω)) |
| (hH : ∀ a, IsStronglyProgressive F (H a)) |
| (hc : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => H a t ω)) |
| {C : ι → ℝ} (hC : ∀ a, 0 ≤ C a) |
| (hb : ∀ a, ∀ᵐ ω ∂P, ∀ t, |H a t ω| ≤ C a) |
| (henergy : ∀ N : ℕ, Summable (fun a => |
| ∫ s in (0 : ℝ)..(N : ℝ), ∫ ω, H a s.toNNReal ω ^ 2 ∂P)) |
| (hI2 : ∀ a t, MemLp (I a t) 2 P) |
| (hYad : StronglyAdapted F Y) (hY2 : ∀ t, MemLp (Y t) 2 P) |
| (hlim : ∀ a (N : ℕ), 0 < N → ∀ t ≤ (N : ℝ≥0), |
| Tendsto (fun n => ∫ ω, |
| (BrownianIntegralProcess.stoppedUniformSum (B a) (H a) N n t ω - |
| I a t ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (hsum : ∀ t, HasSum (fun a => (hI2 a t).toLp (I a t)) |
| ((hY2 t).toLp (Y t))) : |
| Martingale Y F P ∧ |
| Martingale (fun t ω => Y t ω ^ 2 - |
| nnrealTimePrimitive (spatialSquareDensity H) t ω) F P := by |
| obtain ⟨J, Z, hJ2, hZ2, hJ, hJlim, hZp, hZM, hZc, hZ0, |
| hZiso, hZsum, hZmax, hZQ, hZA, hZseries⟩ := |
| exists_global_continuous_spatial_integral_compensatedSquare |
| hB hAd hpast hind hBc (fun a => (hH a).stronglyAdapted) |
| hc hC hb henergy |
| have hIJ (a : ι) (t : ℝ≥0) : I a t =ᵐ[P] J a t := by |
| obtain ⟨N, htN⟩ := exists_nat_gt (t : ℝ) |
| have hN : 0 < N := by exact_mod_cast (t.coe_nonneg.trans_lt htN) |
| have htN' : t ≤ (N : ℝ≥0) := by exact_mod_cast htN.le |
| apply BrownianGrid.ae_eq_of_mean_square_limits |
| (fun n => StoppedIntegralAssociativity.stoppedUniformSum_memLp |
| (fun s => (hB a).isGaussianProcess.hasGaussianLaw_eval s |>.memLp_two) |
| (hH a).stronglyAdapted (hb a) (N : ℝ≥0) n t) |
| (hI2 a t) (hJ2 a t) |
| · exact hlim a N hN t htN' |
| · exact hJlim a N hN t htN' |
| have hYZ (t : ℝ≥0) : Y t =ᵐ[P] Z t := by |
| have heq : (fun a => (hI2 a t).toLp (I a t)) = |
| (fun a => (hJ2 a t).toLp (J a t)) := |
| funext (fun a => MemLp.toLp_congr (hI2 a t) (hJ2 a t) (hIJ a t)) |
| have hs := hsum t |
| rw [heq] at hs |
| exact (MemLp.toLp_eq_toLp_iff (hY2 t) (hZ2 t)).mp (hs.unique (hZsum t)) |
| have hQ := compensatedSquare_spatial_primitive hH hc |
| (fun N => hZseries (N : ℝ≥0)) hZM.stronglyAdapted hZQ |
| refine ⟨hZM.congr hYad (fun t => (hYZ t).symm), ?_⟩ |
| apply hQ.congr (fun t => ((hYad t).pow 2).sub |
| (stronglyAdapted_nnrealTimePrimitive (spatialSquareDensity_isStronglyProgressive hH) t)) |
| intro t |
| filter_upwards [hYZ t] with ω hω |
| exact congrArg (fun z => z ^ 2 - nnrealTimePrimitive (spatialSquareDensity H) t ω) hω.symm |
| |
| theorem difference_compensatedSquare_of_spatial_integral_limits |
| [Countable ι] [IsProbabilityMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {B H K I J : ι → ℝ≥0 → Ω → ℝ} |
| {Y Z : ℝ≥0 → Ω → ℝ} |
| (hB : ∀ a, IsPreBrownianReal (B a) P) |
| (hAd : ∀ a, StronglyAdapted F (B a)) |
| (hpast : ∀ s t, s ≤ t → Indep (MeasurableSpace.comap |
| (fun ω a => B a t ω - B a s ω) inferInstance) (F s) P) |
| (hind : ∀ s t, s ≤ t → Pairwise fun a b => |
| IndepFun (fun ω => B a t ω - B a s ω) |
| (fun ω => B b t ω - B b s ω) P) |
| (hBc : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => B a t ω)) |
| (hH : ∀ a, IsStronglyProgressive F (H a)) |
| (hK : ∀ a, IsStronglyProgressive F (K a)) |
| (hcH : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => H a t ω)) |
| (hcK : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => K a t ω)) |
| {C D : ι → ℝ} (hC : ∀ a, 0 ≤ C a) (hD : ∀ a, 0 ≤ D a) |
| (hbH : ∀ a, ∀ᵐ ω ∂P, ∀ t, |H a t ω| ≤ C a) |
| (hbK : ∀ a, ∀ᵐ ω ∂P, ∀ t, |K a t ω| ≤ D a) |
| (heH : ∀ N : ℕ, Summable (fun a => |
| ∫ s in (0 : ℝ)..(N : ℝ), ∫ ω, H a s.toNNReal ω ^ 2 ∂P)) |
| (heK : ∀ N : ℕ, Summable (fun a => |
| ∫ s in (0 : ℝ)..(N : ℝ), ∫ ω, K a s.toNNReal ω ^ 2 ∂P)) |
| (hI2 : ∀ a t, MemLp (I a t) 2 P) (hJ2 : ∀ a t, MemLp (J a t) 2 P) |
| (hYad : StronglyAdapted F Y) (hZad : StronglyAdapted F Z) |
| (hY2 : ∀ t, MemLp (Y t) 2 P) (hZ2 : ∀ t, MemLp (Z t) 2 P) |
| (hlimI : ∀ a (N : ℕ), 0 < N → ∀ t ≤ (N : ℝ≥0), |
| Tendsto (fun n => ∫ ω, |
| (BrownianIntegralProcess.stoppedUniformSum (B a) (H a) N n t ω - |
| I a t ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (hlimJ : ∀ a (N : ℕ), 0 < N → ∀ t ≤ (N : ℝ≥0), |
| Tendsto (fun n => ∫ ω, |
| (BrownianIntegralProcess.stoppedUniformSum (B a) (K a) N n t ω - |
| J a t ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (hsY : ∀ t, HasSum (fun a => (hI2 a t).toLp (I a t)) |
| ((hY2 t).toLp (Y t))) |
| (hsZ : ∀ t, HasSum (fun a => (hJ2 a t).toLp (J a t)) |
| ((hZ2 t).toLp (Z t))) : |
| Martingale (fun t ω => Y t ω - Z t ω) F P ∧ |
| Martingale (fun t ω => (Y t ω - Z t ω) ^ 2 - |
| nnrealTimePrimitive |
| (spatialSquareDensity (fun a t ω => H a t ω - K a t ω)) t ω) F P := by |
| have hc (a : ι) : ∀ᵐ ω ∂P, Continuous (fun t => H a t ω - K a t ω) := by |
| filter_upwards [hcH a, hcK a] with ω hω kω |
| exact hω.sub kω |
| have hb (a : ι) : ∀ᵐ ω ∂P, ∀ t, |H a t ω - K a t ω| ≤ C a + D a := by |
| filter_upwards [hbH a, hbK a] with ω hω kω |
| exact fun t => (abs_sub _ _).trans (add_le_add (hω t) (kω t)) |
| have he (N : ℕ) : Summable (fun a => ∫ s in (0 : ℝ)..(N : ℝ), |
| ∫ ω, (H a s.toNNReal ω - K a s.toNNReal ω) ^ 2 ∂P) := by |
| by_cases hN : 0 < N |
| · apply summable_difference_energy_of_row_energies hB hAd hpast H K |
| (fun a => (hH a).stronglyAdapted) (fun a => (hK a).stronglyAdapted) |
| hcH hcK C D hC hD hbH hbK (N : ℝ≥0) (heH N) (heK N) |
| (fun a => hI2 a N) (fun a => hJ2 a N) |
| · intro a |
| simpa only [BrownianIntegralProcess.stoppedUniformSum_terminal] using |
| hlimI a N hN N le_rfl |
| · intro a |
| simpa only [BrownianIntegralProcess.stoppedUniformSum_terminal] using |
| hlimJ a N hN N le_rfl |
| · have hn : N = 0 := Nat.eq_zero_of_not_pos hN |
| simp only [hn, Nat.cast_zero, intervalIntegral.integral_same] |
| exact summable_zero |
| apply compensatedSquare_of_spatial_integral_limits hB hAd hpast hind hBc |
| (fun a t => (hH a t).sub (hK a t)) hc (fun a => add_nonneg (hC a) (hD a)) hb he |
| (fun a t => (hI2 a t).sub (hJ2 a t)) |
| (fun t => (hYad t).sub (hZad t)) (fun t => (hY2 t).sub (hZ2 t)) |
| · intro a N hN t htN |
| have hM2 (s : ℝ≥0) := (hB a).isGaussianProcess.hasGaussianLaw_eval s |>.memLp_two |
| have h := SimpleIntegral.meanSquare_limit_sub |
| (fun n => StoppedIntegralAssociativity.stoppedUniformSum_memLp |
| hM2 (hH a).stronglyAdapted (hbH a) N n t) |
| (fun n => StoppedIntegralAssociativity.stoppedUniformSum_memLp |
| hM2 (hK a).stronglyAdapted (hbK a) N n t) |
| (hI2 a t) (hJ2 a t) (hlimI a N hN t htN) (hlimJ a N hN t htN) |
| have heq (n : ℕ) (ω : Ω) : |
| BrownianIntegralProcess.stoppedUniformSum (B a) |
| (fun s ω => H a s ω - K a s ω) N n t ω = |
| BrownianIntegralProcess.stoppedUniformSum (B a) (H a) N n t ω - |
| BrownianIntegralProcess.stoppedUniformSum (B a) (K a) N n t ω := by |
| simp only [BrownianIntegralProcess.stoppedUniformSum, |
| StoppedSimpleIntegral.intervalSum, sub_mul, Finset.sum_sub_distrib] |
| simpa only [heq, Pi.sub_apply] using h |
| · intro t |
| exact (hsY t).sub (hsZ t) |
| |
| end Problem367.BrownianArrayFiltration |
| |
| end |
| |
| end OAI |
| |
