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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 | 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
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