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f61e547 | 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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 | 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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