| import OAI.Probability.FreeIsing.Brownian.ArrayFiltrationIntegral |
| import OAI.Probability.FreeIsing.Brownian.IntegralHorizon |
| import OAI.Probability.FreeIsing.Martingales.SpatialLimit |
|
|
| 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} |
| {B : ι → ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ} |
|
|
| theorem stopped_grid_sums_orthogonal [IsProbabilityMeasure P] |
| (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) |
| (τ : ℕ → ℝ≥0) (hτ : Monotone τ) (H : ι → ℕ → Ω → ℝ) |
| (hH : ∀ a i, StronglyMeasurable[F (τ i)] (H a i)) |
| (hH2 : ∀ a i, MemLp (H a i) 2 P) |
| {a b : ι} (hab : a ≠ b) (u v : Finset ℕ) (t : ℝ≥0) : |
| (∫ ω, |
| (∑ i ∈ u, StoppedSimpleIntegral.intervalSum (B a) (H a i) (τ i) (τ (i + 1)) t ω) * |
| (∑ i ∈ v, StoppedSimpleIntegral.intervalSum (B b) (H b i) (τ i) (τ (i + 1)) t ω) |
| ∂P) = 0 := by |
| classical |
| let σ : ℕ → ℝ≥0 := fun i => min t (τ i) |
| let G : ι → ℕ → Ω → ℝ := fun a i => if τ i ≤ t then H a i else 0 |
| have hσ : Monotone σ := fun i j hij => min_le_min_left t (hτ hij) |
| have hG (a : ι) (i : ℕ) : StronglyMeasurable[F (σ i)] (G a i) := by |
| change StronglyMeasurable[F (min t (τ i))] (if τ i ≤ t then H a i else 0) |
| by_cases hi : τ i ≤ t |
| · rw [ite_eq_left hi, min_eq_right hi] |
| exact hH a i |
| · rw [ite_eq_right hi] |
| exact stronglyMeasurable_zero |
| have hG2 (a : ι) (i : ℕ) : MemLp (G a i) 2 P := by |
| by_cases hi : τ i ≤ t |
| · simpa only [G, ite_eq_left hi] using hH2 a i |
| · simp only [G, ite_eq_right hi] |
| exact MemLp.zero |
| have hterm (a : ι) (i : ℕ) (ω : Ω) : |
| StoppedSimpleIntegral.intervalSum (B a) (H a i) (τ i) (τ (i + 1)) t ω = |
| G a i ω * (B a (σ (i + 1)) ω - B a (σ i) ω) := by |
| by_cases hi : τ i ≤ t |
| · simp only [G, ite_eq_left hi, σ, StoppedSimpleIntegral.intervalSum] |
| · have hti : t ≤ τ i := le_of_not_ge hi |
| have hti' : t ≤ τ (i + 1) := hti.trans (hτ (Nat.le_succ i)) |
| simp only [G, ite_eq_right hi, σ, StoppedSimpleIntegral.intervalSum, |
| min_eq_left hti, min_eq_left hti', sub_self, mul_zero, Pi.zero_apply] |
| simp_rw [hterm] |
| exact grid_sums_orthogonal hB hAd hind hcross σ hσ G hG hG2 hab u v |
|
|
| theorem stopped_uniform_limits_orthogonal [IsProbabilityMeasure P] |
| (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) |
| {H : ι → ℝ≥0 → Ω → ℝ} (hH : ∀ a, StronglyAdapted F (H a)) |
| (hH2 : ∀ a t, MemLp (H a t) 2 P) (T t : ℝ≥0) |
| {I : ι → Ω → ℝ} (hI2 : ∀ a, MemLp (I a) 2 P) |
| (hlim : ∀ a, Tendsto (fun n => ∫ ω, |
| (BrownianIntegralProcess.stoppedUniformSum (B a) (H a) T n t ω - I a ω) ^ 2 ∂P) |
| atTop (𝓝 0)) : |
| Pairwise (fun a b => ∫ ω, I a ω * I b ω ∂P = 0) := by |
| have hi (a : ι) (s t : ℝ≥0) (hst : s ≤ t) : |
| Indep (MeasurableSpace.comap (fun ω => B a t ω - B a s ω) inferInstance) (F s) P := |
| coordinate_increment_indep_past (hind s t hst) a |
| have hS2 (a : ι) (n : ℕ) : |
| MemLp (BrownianIntegralProcess.stoppedUniformSum (B a) (H a) T n t) 2 P := |
| BrownianFiltration.stopped_predictable_sum_memLp_two (hB a) (hi a) |
| (BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n) |
| (fun i => H a (BrownianGrid.uniformGrid T n i)) (fun i => hH a _) (fun i => hH2 a _) |
| (Finset.range n) t |
| have hLp (a : ι) := BrownianIntegralProcess.L2_tendsto_of_integral_sq_sub_tendsto |
| (hS2 a) (hI2 a) (hlim a) |
| intro a b hab |
| have hinner := (hLp a).inner (𝕜 := ℝ) (hLp b) |
| have hz (n : ℕ) : inner ℝ |
| ((hS2 a n).toLp (BrownianIntegralProcess.stoppedUniformSum (B a) (H a) T n t)) |
| ((hS2 b n).toLp (BrownianIntegralProcess.stoppedUniformSum (B b) (H b) T n t)) = 0 := by |
| rw [PredictableL2.inner_toLp] |
| exact stopped_grid_sums_orthogonal hB hAd hind hcross |
| (BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n) |
| (fun a i => H a (BrownianGrid.uniformGrid T n i)) |
| (fun a i => hH a _) (fun a i => hH2 a _) hab (Finset.range n) (Finset.range n) t |
| have heq : inner ℝ ((hI2 a).toLp (I a)) ((hI2 b).toLp (I b)) = 0 := |
| tendsto_nhds_unique hinner (by simpa only [hz] using |
| (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0))) |
| simpa only [PredictableL2.inner_toLp] using heq |
|
|
| theorem exists_global_continuous_spatial_integral [Countable ι] [IsProbabilityMeasure P] |
| (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 : ι → ℝ≥0 → Ω → ℝ) (hH : ∀ a, StronglyAdapted 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)) : |
| ∃ (I : ι → ℝ≥0 → Ω → ℝ) (Y : ℝ≥0 → Ω → ℝ) |
| (hI2 : ∀ a t, MemLp (I a t) 2 P) (hY2 : ∀ t, MemLp (Y t) 2 P), |
| (∀ a, IsStronglyPredictable F (I a) ∧ Martingale (I a) F P ∧ |
| (∀ᵐ ω ∂P, Continuous (fun t => I a t ω))) ∧ |
| (∀ 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)) ∧ |
| IsStronglyPredictable F Y ∧ Martingale Y F P ∧ |
| (∀ᵐ ω ∂P, Continuous (fun t => Y t ω)) ∧ Y 0 =ᵐ[P] 0 ∧ |
| (∀ t, (∫ ω, Y t ω ∂P) = 0) ∧ |
| (∀ t, (∫ ω, Y t ω ^ 2 ∂P) = |
| ∑' a, ∫ s in (0 : ℝ)..(t : ℝ), ∫ ω, H a s.toNNReal ω ^ 2 ∂P) ∧ |
| (∀ t, HasSum (fun a => (hI2 a t).toLp (I a t)) ((hY2 t).toLp (Y t))) ∧ |
| (∀ T : ℝ≥0, Tendsto (fun q : Finset ι => |
| ∫ ω, MartingaleStrongMaximal.maximalSquare |
| (MartingaleSpatialLimit.finiteSum I q - Y) T ω ∂P) atTop (𝓝 0)) := by |
| have hi (a : ι) (s t : ℝ≥0) (hst : s ≤ t) : |
| Indep (MeasurableSpace.comap (fun ω => B a t ω - B a s ω) inferInstance) (F s) P := |
| coordinate_increment_indep_past (hind s t hst) a |
| have hH2 (a : ι) (t : ℝ≥0) : MemLp (H a t) 2 P := by |
| apply MemLp.of_bound ((hH a t).mono (F.le t)).aestronglyMeasurable (C a) |
| filter_upwards [hb a] with ω hω |
| simpa only [Real.norm_eq_abs] using hω t |
| have hex (a : ι) := BrownianIntegralProcess.exists_global_continuous_uniform_integral |
| (hB a) (hAd a) (hi a) (hBc a) (hH a) (hc a) (hC a) (hb a) |
| choose I hIp hIM hIc hI2 hIz hImean hIiso hIlim using hex |
| have horth (t : ℝ≥0) : Pairwise (fun a b => ∫ ω, I a t ω * I b t ω ∂P = 0) := by |
| obtain ⟨N, hN⟩ := exists_nat_ge t |
| have ht : t ≤ ((N + 1 : ℕ) : ℝ≥0) := |
| hN.trans (by exact_mod_cast Nat.le_succ N) |
| exact stopped_uniform_limits_orthogonal hB hAd hind hcross hH hH2 |
| ((N + 1 : ℕ) : ℝ≥0) t (fun a => hI2 a t) |
| (fun a => hIlim a (N + 1) (Nat.succ_pos N) t ht) |
| have hIenergy (N : ℕ) : Summable (fun a => ∫ ω, I a (N : ℝ≥0) ω ^ 2 ∂P) := by |
| simpa only [hIiso, NNReal.coe_natCast] using henergy N |
| obtain ⟨Y, hY2, hYp, hYM, hYc, hYsum, hYmax⟩ := |
| MartingaleSpatialLimit.exists_continuous_martingale_sum hIM hI2 hIc horth hIenergy |
| have hYiso (t : ℝ≥0) : (∫ ω, Y t ω ^ 2 ∂P) = |
| ∑' a, ∫ s in (0 : ℝ)..(t : ℝ), ∫ ω, H a s.toNNReal ω ^ 2 ∂P := by |
| have he := MartingaleSpatialLimit.summable_energy_of_integer hIM hI2 hIenergy t |
| have hinner : Pairwise (fun a b => inner ℝ |
| ((hI2 a t).toLp (I a t)) ((hI2 b t).toLp (I b t)) = 0) := by |
| intro a b hab |
| simpa only [PredictableL2.inner_toLp] using horth t hab |
| have hn := PredictableL2.norm_tsum_sq_of_orthogonal |
| (fun a => (hI2 a t).toLp (I a t)) hinner |
| (by simpa only [PredictableL2.norm_toLp_sq] using he) |
| rw [(hYsum t).tsum_eq] at hn |
| simpa only [PredictableL2.norm_toLp_sq, hIiso] using hn |
| have hYzero : Y 0 =ᵐ[P] 0 := by |
| apply ae_eq_of_integral_sq_sub_eq_zero |
| · simpa only [Pi.zero_apply, sub_zero] using (hY2 0).integrable_sq |
| · simpa only [Pi.zero_apply, sub_zero, NNReal.coe_zero, |
| intervalIntegral.integral_same, tsum_zero] using hYiso 0 |
| exact ⟨I, Y, hI2, hY2, (fun a => ⟨hIp a, hIM a, hIc a⟩), hIlim, |
| hYp, hYM, hYc, hYzero, |
| (fun t => PredictableL2.integral_eq_zero_of_hasSum |
| (fun a => I a t) (fun a => hI2 a t) (hY2 t) (hYsum t) (fun a => hImean a t)), |
| hYiso, hYsum, hYmax⟩ |
|
|
| end Problem367.BrownianArrayFiltration |
|
|
| end |
|
|
| end OAI |
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|
