File size: 7,855 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 | import OAI.Probability.FreeIsing.Brownian.ArrayFiltrationIntegral
import OAI.Probability.FreeIsing.Brownian.IntegralProcess
import OAI.Probability.FreeIsing.Estimates.CompensatedSquareLimit
import OAI.Probability.FreeIsing.Stochastic.SelectedCompensator
import OAI.Probability.FreeIsing.Stochastic.StoppedBrownianArraySquare
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal ENNReal Topology BigOperators
namespace Problem367.BrownianArrayCompensator
variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Fintype ι]
{P : Measure Ω} [IsProbabilityMeasure P] {F : Filtration ℝ≥0 mΩ}
theorem compensatedSquare_of_terminal_meanSquare
{B H : ι → ℝ≥0 → Ω → ℝ} {M : ℝ≥0 → Ω → ℝ}
(hB : ∀ j, IsPreBrownianReal (B j) P)
(hAd : ∀ j, StronglyAdapted F (B j))
(hpast : ∀ s t, s ≤ t → Indep
(MeasurableSpace.comap (fun ω j => B j t ω - B j s ω) inferInstance) (F s) P)
(hcross : ∀ s t, s ≤ t → Pairwise fun j k =>
IndepFun (fun ω => B j t ω - B j s ω) (fun ω => B k t ω - B k s ω) P)
(hH : ∀ j, StronglyAdapted F (H j))
(hc : ∀ j, ∀ᵐ ω ∂P, Continuous (fun t => H j t ω))
(C : ι → ℝ) (hC : ∀ j, 0 ≤ C j)
(hb : ∀ j, ∀ᵐ ω ∂P, ∀ t, |H j t ω| ≤ C j)
(hM : Martingale M F P) (hM2 : ∀ t, MemLp (M t) 2 P)
(hlim : ∀ T, Tendsto (fun n : ℕ => ∫ ω,
((∑ j, BrownianGrid.uniformSum (B j) (H j) T n ω) - M T ω) ^ 2 ∂P)
atTop (𝓝 0)) :
Martingale (fun t ω => M t ω ^ 2 -
∑ j, SelectedCompensator.selected (H j) t t ω) F P := by
classical
have hi (j : ι) (s t : ℝ≥0) (hst : s ≤ t) :=
BrownianArrayFiltration.coordinate_increment_indep_past (hpast s t hst) j
have hH2 (j : ι) (t : ℝ≥0) : MemLp (H j t) 2 P := by
apply MemLp.of_bound (hH j).stronglyMeasurable.aestronglyMeasurable (C j)
filter_upwards [hb j] with ω hω
simpa only [Real.norm_eq_abs] using hω t
have hU2 (T : ℝ≥0) (n : ℕ) (j : ι) :
MemLp (BrownianGrid.uniformSum (B j) (H j) T n) 2 P :=
BrownianFiltrationGrid.uniformSum_memLp (hB j) (hAd j) (hi j)
(H j) (hH j) (hH2 j) T n
have hQ : StronglyAdapted F (fun t ω =>
∑ j, SelectedCompensator.selected (H j) t t ω) := by
intro t
exact Finset.stronglyMeasurable_fun_sum _ fun j _ =>
SelectedCompensator.selected_stronglyAdapted (hH j) t t
refine ⟨fun t => ((hM.stronglyAdapted t).pow 2).sub (hQ t), ?_⟩
intro s T hsT
let U (n : ℕ) : Ω → ℝ := fun ω =>
∑ j, BrownianGrid.uniformSum (B j) (H j) T n ω
let X (n : ℕ) (t : ℝ≥0) (ω : Ω) : ℝ :=
∑ j, BrownianIntegralProcess.stoppedUniformSum (B j) (H j) T n t ω
let A (n : ℕ) (t : ℝ≥0) (ω : Ω) : ℝ :=
∑ j, SelectedCompensator.squareSum (H j) T n t ω
let Y (t : ℝ≥0) : Ω → ℝ := P[M T | F t]
let Q (t : ℝ≥0) (ω : Ω) : ℝ :=
∑ j, SelectedCompensator.selected (H j) T t ω
have hu2 (n : ℕ) : MemLp (U n) 2 P :=
memLp_finsetSum _ fun j _ => hU2 T n j
have hx2 (n : ℕ) (t : ℝ≥0) : MemLp (X n t) 2 P := by
apply memLp_finsetSum
intro j hj
exact BrownianFiltration.stopped_predictable_sum_memLp_two (hB j) (hi j)
(BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n)
(fun i => H j (BrownianGrid.uniformGrid T n i)) (fun i => hH j _)
(fun i => hH2 j _) (Finset.range n) t
have ha (n : ℕ) (t : ℝ≥0) : Integrable (A n t) P :=
integrable_finsetSum _ fun j _ =>
(SelectedCompensator.squareSum_memLp (hH j) (hC j) (hb j) T t n 1).integrable le_rfl
have hq (t : ℝ≥0) : Integrable (Q t) P :=
integrable_finsetSum _ fun j _ =>
(SelectedCompensator.selected_memLp (hH j) (hc j) (hC j) (hb j) T t 1).integrable le_rfl
have hy2 (t : ℝ≥0) : MemLp (Y t) 2 P := (hM2 T).condExp (by norm_num)
have hce (n : ℕ) (t : ℝ≥0) : P[U n | F t] =ᵐ[P] X n t := by
have hsum := condExp_finsetSum (s := Finset.univ)
(fun j _ => (hU2 T n j).integrable (by norm_num)) (F t)
have heach : ∀ᵐ ω ∂P, ∀ j,
P[BrownianGrid.uniformSum (B j) (H j) T n | F t] ω =
BrownianIntegralProcess.stoppedUniformSum (B j) (H j) T n t ω :=
ae_all_iff.mpr fun j => BrownianIntegralProcess.stoppedUniformSum_eq_condExp
(hB j) (hAd j) (hi j) (hH j) (hb j) T n t
have hsumFun : (∑ j, BrownianGrid.uniformSum (B j) (H j) T n) = U n := by
funext ω
simp only [U, Finset.sum_apply]
rw [hsumFun] at hsum
filter_upwards [hsum, heach] with ω hω he
simp only [Finset.sum_apply] at hω
exact hω.trans (Finset.sum_congr rfl (fun j _ => he j))
have hxlim (t : ℝ≥0) : Tendsto (fun n => ∫ ω,
(X n t ω - Y t ω) ^ 2 ∂P) atTop (𝓝 0) := by
apply squeeze_zero (fun n => integral_nonneg (fun ω => sq_nonneg _)) ?_ (hlim T)
intro n
calc
_ = ∫ ω, (P[U n | F t] ω - P[M T | F t] ω) ^ 2 ∂P := by
apply integral_congr_ae
filter_upwards [hce n t] with ω hω
exact congrArg (fun z => (z - Y t ω) ^ 2) hω.symm
_ ≤ _ := BrownianIntegralProcess.meanSquare_condExp_sub_le (F t) (hu2 n) (hM2 T)
have hfinite (n : ℕ) : Martingale (fun t ω => (X n t ω) ^ 2 - A n t ω) F P := by
have hh := StoppedBrownianArraySquare.stopped_brownian_array_square_martingale
B hB hAd hpast hcross (BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n)
(fun j i => H j (BrownianGrid.uniformGrid T n i)) (fun j i => hH j _)
(fun j _ => C j)
(fun j i => (hb j).mono (fun ω hω => by simpa only [Real.norm_eq_abs] using hω _))
Finset.univ n
simpa only [X, A, BrownianIntegralProcess.stoppedUniformSum,
SelectedCompensator.squareSum, mul_comm] using hh
have halim (t : ℝ≥0) : Tendsto (fun n => eLpNorm (A n t - Q t) 1 P) atTop (𝓝 0) := by
have hl := tendsto_finsetSum Finset.univ (fun j _ =>
SelectedCompensator.tendsto_squareSum_L1_selected
(fun r => (hH j).stronglyMeasurable.aestronglyMeasurable)
(hc j) (hC j) (hb j) T t)
simp only [Finset.sum_const_zero] at hl
apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hl (fun _ => bot_le)
intro n
change eLpNorm (A n t - Q t) 1 P ≤ ∑ j,
eLpNorm (SelectedCompensator.squareSum (H j) T n t - SelectedCompensator.selected (H j) T t) 1 P
have heq : A n t - Q t = ∑ j,
(SelectedCompensator.squareSum (H j) T n t - SelectedCompensator.selected (H j) T t) := by
funext ω
simp only [A, Q, Pi.sub_apply, Finset.sum_apply, Finset.sum_sub_distrib]
rw [heq]
exact eLpNorm_sum_le le_rfl
have hcomp := CompensatedSquareLimit.martingale_compensatedSquare_of_limits
hx2 ha hfinite (martingale_condExp (M T) F P).stronglyAdapted hy2
(fun t => Finset.stronglyMeasurable_fun_sum _ fun j _ =>
SelectedCompensator.selected_stronglyAdapted (hH j) T t) hq
(fun t => CompensatedSquareLimit.tendsto_eLpNorm_two_of_meanSquare
(fun n => hx2 n t) (hy2 t) (hxlim t)) halim
have heq (r : ℝ≥0) (hr : r ≤ T) :
(fun ω => (Y r ω) ^ 2 - Q r ω) =ᵐ[P]
(fun ω => M r ω ^ 2 - ∑ j, SelectedCompensator.selected (H j) r r ω) := by
have hcAll : ∀ᵐ ω ∂P, ∀ j, Continuous (fun t => H j t ω) := ae_all_iff.mpr hc
filter_upwards [hM.condExp_ae_eq hr, hcAll] with ω hω hcω
change (P[M T | F r] ω) ^ 2 - _ = _
rw [hω]
congr 1
apply Finset.sum_congr rfl
intro j hj
simp only [SelectedCompensator.selected_eq_integral_of_continuous (hcω j),
min_self, min_eq_left (show (r : ℝ) ≤ T from hr)]
exact (condExp_congr_ae (heq T le_rfl).symm).trans
((hcomp.condExp_ae_eq hsT).trans (heq s hsT))
end Problem367.BrownianArrayCompensator
end
end OAI
|