| import OAI.Probability.FreeIsing.Brownian.ContinuousSpatialIntegral |
| import OAI.Probability.FreeIsing.Brownian.RowEnergyCompletion |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology BigOperators |
|
|
| namespace Problem367.MartingaleSquareMaximal |
|
|
| theorem exists_global_continuous_meanSquare_limit |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {M : ℕ → ℝ≥0 → Ω → ℝ} |
| (hM : ∀ n, Martingale (M n) F P) |
| (h2 : ∀ n t, MemLp (M n t) 2 P) |
| (hc : ∀ n, ∀ᵐ ω ∂P, Continuous (fun t => M n t ω)) |
| (hC : ∀ N : ℕ, 0 < N → ∀ ε > (0 : ℝ), ∃ k : ℕ, |
| ∀ n ≥ k, ∀ m ≥ k, (∫ ω, (M n N ω - M m N ω) ^ 2 ∂P) < ε) : |
| ∃ (Z : ℝ≥0 → Ω → ℝ) (_hZ2 : ∀ t, MemLp (Z t) 2 P), |
| IsStronglyPredictable F Z ∧ Martingale Z F P ∧ |
| (∀ᵐ ω ∂P, Continuous (fun t => Z t ω)) ∧ |
| (∀ t, Tendsto (fun n => ∫ ω, (M n t ω - Z t ω) ^ 2 ∂P) atTop (𝓝 0)) ∧ |
| (∀ t, Tendsto (fun n => ∫ ω, M n t ω ^ 2 ∂P) atTop |
| (𝓝 (∫ ω, Z t ω ^ 2 ∂P))) ∧ |
| (∀ T, Tendsto (fun n => ∫ ω, |
| MartingaleStrongMaximal.maximalSquare (M n - Z) T ω ∂P) atTop (𝓝 0)) := by |
| have hCpos (N : ℕ) (hN : 0 < N) : |
| CauchySeq (fun n => (h2 n (N : ℝ≥0)).toLp (M n N)) := by |
| apply Metric.cauchySeq_iff.mpr |
| intro ε hε |
| obtain ⟨k, hk⟩ := hC N hN (ε ^ 2) (sq_pos_of_pos hε) |
| refine ⟨k, fun n hn m hm => ?_⟩ |
| have hs := hk n hn m hm |
| rw [← dist_toLp_sq (h2 n N) (h2 m N)] at hs |
| nlinarith [dist_nonneg (x := (h2 n N).toLp (M n N)) |
| (y := (h2 m N).toLp (M m N))] |
| have hCall (N : ℕ) : CauchySeq (fun n => (h2 n (N : ℝ≥0)).toLp (M n N)) := |
| L2_cauchy_of_le hM h2 (by exact_mod_cast Nat.le_succ N) |
| (hCpos (N + 1) (Nat.succ_pos N)) |
| obtain ⟨Z, hpZ, hcZ, hlZ⟩ := |
| exists_global_predictable_continuous_L2_limit hM h2 hc hCall |
| choose hZ2 hl using hlZ |
| have hmartZ : Martingale Z F P := |
| martingale_of_L2_limits hM h2 hpZ.stronglyAdapted hZ2 hl |
| have hmean (t : ℝ≥0) := BrownianIntegralProcess.integral_sq_sub_tendsto_of_L2_tendsto |
| (fun n => h2 n t) (hZ2 t) (hl t) |
| refine ⟨Z, hZ2, hpZ, hmartZ, hcZ, hmean, ?_, ?_⟩ |
| · intro t |
| simpa only [PredictableL2.norm_toLp_sq] using (hl t).norm.pow 2 |
| · intro T |
| apply squeeze_zero (fun n => integral_nonneg (fun ω => ENNReal.toReal_nonneg)) |
| (fun n => ?_) (show Tendsto |
| (fun n => 4 * ∫ ω, (M n T ω - Z T ω) ^ 2 ∂P) atTop (𝓝 0) from by |
| simpa only [mul_zero] using (hmean T).const_mul 4) |
| exact MartingaleStrongMaximal.integral_maximalSquare_le |
| ((hM n).sub hmartZ) (fun t => (h2 n t).sub (hZ2 t)) |
| (by filter_upwards [hc n, hcZ] with ω hn hz; exact hn.sub hz) T |
|
|
| end Problem367.MartingaleSquareMaximal |
|
|
| namespace Problem367.BrownianArrayFiltration |
|
|
| theorem exists_global_continuous_integral_limit_of_row_energy_cauchy |
| {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Countable ι] |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| {B : ι → ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ} |
| (hB : ∀ a, IsPreBrownianReal (B a) P) |
| (hAd : ∀ a, StronglyAdapted F (B a)) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω a => B a t ω - B a s ω) inferInstance) (F s) P) |
| (hcross : ∀ 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)) |
| (H : ℕ → ι → ℝ≥0 → Ω → ℝ) |
| (hH : ∀ n a, StronglyAdapted F (H n a)) |
| (hcH : ∀ n a, ∀ᵐ ω ∂P, Continuous (fun t => H n a t ω)) |
| (C : ℕ → ι → ℝ) (hC : ∀ n a, 0 ≤ C n a) |
| (hbH : ∀ n a, ∀ᵐ ω ∂P, ∀ t, |H n a t ω| ≤ C n a) |
| (he : ∀ n (N : ℕ), Summable fun a => ∫ t in (0 : ℝ)..(N : ℝ), |
| ∫ ω, H n a t.toNNReal ω ^ 2 ∂P) |
| (hCauchy : ∀ N : ℕ, 0 < N → ∀ ε > (0 : ℝ), ∃ k : ℕ, |
| ∀ n ≥ k, ∀ m ≥ k, (∑' a, ∫ t in (0 : ℝ)..(N : ℝ), |
| ∫ ω, (H n a t.toNNReal ω - H m a t.toNNReal ω) ^ 2 ∂P) < ε) : |
| ∃ (I : ℕ → ι → ℝ≥0 → Ω → ℝ) (Y : ℕ → ℝ≥0 → Ω → ℝ) |
| (Z : ℝ≥0 → Ω → ℝ) |
| (hI2 : ∀ n a t, MemLp (I n a t) 2 P) |
| (hY2 : ∀ n t, MemLp (Y n t) 2 P) (_hZ2 : ∀ t, MemLp (Z t) 2 P), |
| (∀ n, IsStronglyPredictable F (Y n) ∧ Martingale (Y n) F P ∧ |
| (∀ᵐ ω ∂P, Continuous (fun t => Y n t ω))) ∧ |
| (∀ n a (N : ℕ), 0 < N → Tendsto (fun k => ∫ ω, |
| (BrownianGrid.uniformSum (B a) (H n a) N k ω - I n a N ω) ^ 2 ∂P) |
| atTop (𝓝 0)) ∧ |
| (∀ n t, HasSum (fun a => (hI2 n a t).toLp (I n a t)) |
| ((hY2 n t).toLp (Y n t))) ∧ |
| (∀ n t, (∫ ω, Y n t ω ^ 2 ∂P) = |
| ∑' a, ∫ s in (0 : ℝ)..(t : ℝ), ∫ ω, H n a s.toNNReal ω ^ 2 ∂P) ∧ |
| (∀ n m (N : ℕ), 0 < N → (∫ ω, (Y n N ω - Y m N ω) ^ 2 ∂P) = |
| ∑' a, ∫ s in (0 : ℝ)..(N : ℝ), |
| ∫ ω, (H n a s.toNNReal ω - H m a s.toNNReal ω) ^ 2 ∂P) ∧ |
| IsStronglyPredictable F Z ∧ Martingale Z F P ∧ |
| (∀ᵐ ω ∂P, Continuous (fun t => Z t ω)) ∧ Z 0 =ᵐ[P] 0 ∧ |
| (∀ t, (∫ ω, Z t ω ∂P) = 0) ∧ |
| (∀ t, Tendsto (fun n => ∫ ω, (Y n t ω - Z t ω) ^ 2 ∂P) atTop (𝓝 0)) ∧ |
| (∀ t : ℝ≥0, Tendsto (fun n => ∑' a, ∫ s in (0 : ℝ)..(t : ℝ), |
| ∫ ω, H n a s.toNNReal ω ^ 2 ∂P) atTop (𝓝 (∫ ω, Z t ω ^ 2 ∂P))) ∧ |
| (∀ T, Tendsto (fun n => ∫ ω, |
| MartingaleStrongMaximal.maximalSquare (Y n - Z) T ω ∂P) atTop (𝓝 0)) := by |
| have hex (n : ℕ) := exists_global_continuous_spatial_integral |
| hB hAd hind hcross hBc (H n) (hH n) (hcH n) (C n) (hC n) (hbH n) (he n) |
| choose I Y hI2 hY2 hIprop haI hpY hmY hcY hzY hmeanY heY hsY hmaxY using hex |
| have ha (n : ℕ) (a : ι) (N : ℕ) (hN : 0 < N) : |
| Tendsto (fun k => ∫ ω, |
| (BrownianGrid.uniformSum (B a) (H n a) N k ω - I n a N ω) ^ 2 ∂P) |
| atTop (𝓝 0) := by |
| simpa only [BrownianIntegralProcess.stoppedUniformSum_terminal] using |
| haI n a N hN N le_rfl |
| have hs (n : ℕ) (t : ℝ≥0) : Tendsto (fun s : Finset ι => ∫ ω, |
| ((∑ a ∈ s, I n a t ω) - Y n t ω) ^ 2 ∂P) atTop (𝓝 0) := |
| PredictableL2.tendsto_integral_sq_error_of_hasSum |
| (fun a => I n a t) (fun a => hI2 n a t) (hY2 n t) (hsY n t) |
| have hpair (n m N : ℕ) (hN : 0 < N) : |
| (∫ ω, (Y n N ω - Y m N ω) ^ 2 ∂P) = |
| ∑' a, ∫ s in (0 : ℝ)..(N : ℝ), |
| ∫ ω, (H n a s.toNNReal ω - H m a s.toNNReal ω) ^ 2 ∂P := |
| infinite_row_difference_isometry_of_row_energies hB hAd hind |
| (H n) (H m) (hH n) (hH m) (hcH n) (hcH m) |
| (C n) (C m) (hC n) (hC m) (hbH n) (hbH m) N (he n N) (he m N) |
| hcross (fun a => hI2 n a N) (fun a => hI2 m a N) |
| (fun a => ha n a N hN) (fun a => ha m a N hN) |
| (hY2 n N) (hY2 m N) (hs n N) (hs m N) |
| have hCY (N : ℕ) (hN : 0 < N) : ∀ ε > (0 : ℝ), ∃ k : ℕ, |
| ∀ n ≥ k, ∀ m ≥ k, (∫ ω, (Y n N ω - Y m N ω) ^ 2 ∂P) < ε := by |
| simpa only [hpair _ _ N hN] using hCauchy N hN |
| obtain ⟨Z, hZ2, hpZ, hmZ, hcZ, hlZ, heZ, hmaxZ⟩ := |
| MartingaleSquareMaximal.exists_global_continuous_meanSquare_limit hmY hY2 hcY hCY |
| have hzZ : Z 0 =ᵐ[P] 0 := by |
| have hzero (n : ℕ) : (∫ ω, Y n 0 ω ^ 2 ∂P) = 0 := by |
| simp only [heY, NNReal.coe_zero, intervalIntegral.integral_same, tsum_zero] |
| have henergy0 : (∫ ω, Z 0 ω ^ 2 ∂P) = 0 := |
| tendsto_nhds_unique (heZ 0) (by simpa only [hzero] using |
| (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))) |
| apply ae_eq_of_integral_sq_sub_eq_zero |
| · simpa only [Pi.zero_apply, sub_zero] using (hZ2 0).integrable_sq |
| · simpa only [Pi.zero_apply, sub_zero] using henergy0 |
| have hcenter (t : ℝ≥0) : (∫ ω, Z t ω ∂P) = 0 := by |
| have hlim := BrownianIntegralProcess.L2_tendsto_of_integral_sq_sub_tendsto |
| (fun n => hY2 n t) (hZ2 t) (hlZ t) |
| have hi := (tendsto_const_nhds (x := (memLp_const (1 : ℝ)).toLp |
| (fun _ : Ω => (1 : ℝ)))).inner (𝕜 := ℝ) hlim |
| simp only [PredictableL2.inner_toLp, one_mul, hmeanY] at hi |
| exact tendsto_nhds_unique hi tendsto_const_nhds |
| refine ⟨I, Y, Z, hI2, hY2, hZ2, (fun n => ⟨hpY n, hmY n, hcY n⟩), ha, |
| hsY, heY, hpair, hpZ, hmZ, hcZ, hzZ, hcenter, hlZ, ?_, hmaxZ⟩ |
| intro t |
| simpa only [heY] using heZ t |
| |
| end Problem367.BrownianArrayFiltration |
| |
| end |
| |
| end OAI |
| |
