File size: 8,411 Bytes
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 | 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
|