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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology BigOperators
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 coordinate_increment_indep_past {s t : ℝ≥0}
(hind : Indep
(MeasurableSpace.comap (fun ω a => B a t ω - B a s ω) inferInstance) (F s) P)
(a : ι) :
Indep (MeasurableSpace.comap (fun ω => B a t ω - B a s ω) inferInstance)
(F s) P := by
have hm : Measurable[
MeasurableSpace.comap (fun ω a => B a t ω - B a s ω) inferInstance]
(fun ω => B a t ω - B a s ω) :=
(measurable_pi_apply a).comp (show Measurable[
MeasurableSpace.comap (fun ω a => B a t ω - B a s ω) inferInstance]
(fun ω a => B a t ω - B a s ω) from measurable_iff_comap_le.mpr le_rfl)
exact indep_of_indep_of_le_left hind hm.comap_le
theorem coefficients_indep_increments {s t : ℝ≥0}
(hind : Indep
(MeasurableSpace.comap (fun ω a => B a t ω - B a s ω) inferInstance) (F s) P)
{H : ι → Ω → ℝ} (hH : ∀ a, StronglyMeasurable[F s] (H a)) :
IndepFun (fun ω a => H a ω) (fun ω a => B a t ω - B a s ω) P := by
have hm : Measurable[F s] (fun ω a => H a ω) := by
let : MeasurableSpace Ω := F s
exact Measurable.of_eval fun a => (hH a).measurable
exact indep_of_indep_of_le_left hind.symm hm.comap_le
theorem grid_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)
(t : ℕ → ℝ≥0) (ht : Monotone t) (H : ι → ℕ → Ω → ℝ)
(hH : ∀ a k, StronglyMeasurable[F (t k)] (H a k))
(hH2 : ∀ a k, MemLp (H a k) 2 P) :
Pairwise fun p q : ι × ℕ => ∫ ω,
(H p.1 p.2 ω * (B p.1 (t (p.2 + 1)) ω - B p.1 (t p.2) ω)) *
(H q.1 q.2 ω * (B q.1 (t (q.2 + 1)) ω - B q.1 (t q.2) ω)) ∂P = 0 := by
let G : Filtration ℕ mΩ :=
⟨fun k => F (t k), fun _ _ h => F.mono (ht h), fun k => F.le (t k)⟩
let D : ι → ℕ → Ω → ℝ := fun a k ω => B a (t (k + 1)) ω - B a (t k) ω
have hD2 (a : ι) (k : ℕ) : MemLp (D a k) 2 P :=
((hB a).isGaussianProcess.hasGaussianLaw_eval (t (k + 1))).memLp_two.sub
((hB a).isGaussianProcess.hasGaussianLaw_eval (t k)).memLp_two
have hi (k : ℕ) : IndepFun (fun ω a => H a k ω) (fun ω a => D a k ω) P :=
coefficients_indep_increments (hind _ _ (ht (Nat.le_succ k))) (fun a => hH a k)
have hHD (a : ι) (k : ℕ) : MemLp (fun ω => H a k ω * D a k ω) 2 P :=
PredictableL2.memLp_two_mul_of_indep
((hi k).comp (measurable_pi_apply a) (measurable_pi_apply a)) (hH2 a k) (hD2 a k)
have hz (a : ι) (k : ℕ) : (∫ ω, D a k ω ∂P) = 0 := by
change (∫ ω, B a (t (k + 1)) ω - B a (t k) ω ∂P) = 0
rw [integral_sub ((hB a).integrable_eval _) ((hB a).integrable_eval _),
(hB a).integral_eval, (hB a).integral_eval, sub_self]
have hnoise (k : ℕ) : Pairwise fun a b => ∫ ω, D a k ω * D b k ω ∂P = 0 := by
intro a b hab
have heq := (hcross _ _ (ht (Nat.le_succ k)) hab).integral_fun_mul_eq_mul_integral
(hD2 a k).aestronglyMeasurable (hD2 b k).aestronglyMeasurable
change (∫ ω, D a k ω * D b k ω ∂P) =
(∫ ω, D a k ω ∂P) * ∫ ω, D b k ω ∂P at heq
simpa only [hz, zero_mul] using heq
apply SimpleIntegral.predictable_array_increments_orthogonal G H D hH
· intro a k
exact (hAd a (t (k + 1))).sub
((hAd a (t k)).mono (F.mono (ht (Nat.le_succ k))))
· exact fun a k => (hD2 a k).integrable (by norm_num)
· exact hHD
· intro a k
exact BrownianFiltration.increment_condExp_zero (hB a) (hAd a)
(coordinate_increment_indep_past (hind _ _ (ht (Nat.le_succ k))) a)
· intro k
exact PredictableL2.weighted_array_orthogonal (hi k) (fun a => hH2 a k)
(fun a => hD2 a k) (hnoise k)
theorem 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)
(t : ℕ → ℝ≥0) (ht : Monotone t) (H : ι → ℕ → Ω → ℝ)
(hH : ∀ a k, StronglyMeasurable[F (t k)] (H a k))
(hH2 : ∀ a k, MemLp (H a k) 2 P)
{a b : ι} (hab : a ≠ b) (u v : Finset ℕ) :
(∫ ω, (∑ k ∈ u, H a k ω * (B a (t (k + 1)) ω - B a (t k) ω)) *
(∑ k ∈ v, H b k ω * (B b (t (k + 1)) ω - B b (t k) ω)) ∂P) = 0 := by
have hHD (a : ι) (k : ℕ) : MemLp
(fun ω => H a k ω * (B a (t (k + 1)) ω - B a (t k) ω)) 2 P :=
BrownianFiltration.predictable_mul_memLp_two (hB a)
(coordinate_increment_indep_past (hind _ _ (ht (Nat.le_succ k))) a)
(hH a k) (hH2 a k)
apply SimpleIntegral.integral_mul_sums_eq_zero
(fun k ω => H a k ω * (B a (t (k + 1)) ω - B a (t k) ω))
(fun k ω => H b k ω * (B b (t (k + 1)) ω - B b (t k) ω))
(hHD a) (hHD b) _ u v
intro k l
exact grid_orthogonal hB hAd hind hcross t ht H hH hH2
(show (a, k) ≠ (b, l) from fun h => hab (congrArg Prod.fst h))
theorem exists_channel_integrals [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))
(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) (T : ℝ≥0) :
∃ I : ι → Ω → ℝ,
(∀ a, MemLp (I a) 2 P) ∧
(∀ a, StronglyMeasurable (I a)) ∧
(∀ a, AEStronglyMeasurable[F T] (I a) P) ∧
(∀ a, (∫ ω, I a ω ∂P) = 0) ∧
(∀ a, (∫ ω, I a ω ^ 2 ∂P) =
∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, H a t.toNNReal ω ^ 2 ∂P) ∧
(Pairwise fun a b => ∫ ω, I a ω * I b ω ∂P = 0) ∧
(∀ a, Tendsto (fun n => ∫ ω,
(BrownianGrid.uniformSum (B a) (H a) T n ω - I a ω) ^ 2 ∂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 hmH (a : ι) (t : ℝ≥0) : AEStronglyMeasurable (H a t) P :=
((hH a t).mono (F.le t)).aestronglyMeasurable
have hH2 (a : ι) (t : ℝ≥0) : MemLp (H a t) 2 P := by
apply MemLp.of_bound (hmH a t) (C a)
filter_upwards [hb a] with ω hω
simpa only [Real.norm_eq_abs] using hω t
have hex (a : ι) := BrownianFiltrationGrid.exists_uniformSumLp_limit
(hB a) (hAd a) (hi a) (H a) (hH a) (hH2 a) T
(fun _ hε => MeanSquareContinuity.uniform_integral_sq_sub_le
(hmH a) (hc a) (hC a) (hb a) T hε)
choose I hI using hex
have hprop (a : ι) := BrownianFiltrationGrid.uniformSumLp_limit_terminal_centered
(hB a) (hAd a) (hi a) (H a) (hH a) (hH2 a) T (hI a)
refine ⟨fun a => I a, fun a => Lp.memLp (I a), fun a => Lp.stronglyMeasurable (I a),
fun a => (hprop a).1, fun a => (hprop a).2, ?_, ?_, ?_⟩
· intro a
exact BrownianFiltrationGrid.uniformSumLp_limit_isometry
(hB a) (hAd a) (hi a) (H a) (hH a) (hH2 a) (hc a) (hC a) (hb a) T (hI a)
· intro a b hab
have hlim := (hI a).inner (𝕜 := ℝ) (hI b)
have hz (n : ℕ) : inner ℝ
(BrownianFiltrationGrid.uniformSumLp (hB a) (hAd a) (hi a) (H a) (hH a) (hH2 a) T n)
(BrownianFiltrationGrid.uniformSumLp (hB b) (hAd b) (hi b) (H b) (hH b) (hH2 b) T n) = 0 := by
rw [BrownianFiltrationGrid.uniformSumLp, BrownianFiltrationGrid.uniformSumLp,
PredictableL2.inner_toLp]
exact grid_sums_orthogonal hB hAd hind hcross
(BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n)
(fun a k => H a (BrownianGrid.uniformGrid T n k))
(fun a k => hH a _) (fun a k => hH2 a _) hab (Finset.range n) (Finset.range n)
have heq : inner ℝ (I a) (I b) = 0 :=
tendsto_nhds_unique hlim (by simpa only [hz] using
(tendsto_const_nhds : Tendsto (fun _ : ℕ => (0 : ℝ)) atTop (𝓝 0)))
rw [L2.inner_def] at heq
simpa only [Real.inner_apply] using heq
· intro a
exact BrownianFiltrationGrid.integral_sq_error_tendsto_of_Lp_tendsto
(hB a) (hAd a) (hi a) (H a) (hH a) (hH2 a) T (hI a)
theorem exists_infinite_integral [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))
(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) (T : ℝ≥0)
(henergy : Summable fun a =>
∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, H a t.toNNReal ω ^ 2 ∂P) :
∃ (I : ι → Ω → ℝ) (Y : Ω → ℝ),
(∀ a, MemLp (I a) 2 P) ∧
(∀ a, Tendsto (fun n => ∫ ω,
(BrownianGrid.uniformSum (B a) (H a) T n ω - I a ω) ^ 2 ∂P) atTop (𝓝 0)) ∧
Measurable Y ∧ AEStronglyMeasurable[F T] Y P ∧ MemLp Y 2 P ∧
(∫ ω, Y ω ∂P) = 0 ∧
(∫ ω, Y ω ^ 2 ∂P) =
∑' a, ∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, H a t.toNNReal ω ^ 2 ∂P ∧
Tendsto (fun s : Finset ι => ∫ ω,
((∑ a ∈ s, I a ω) - Y ω) ^ 2 ∂P) atTop (𝓝 0) := by
obtain ⟨I, hI2, _, hIterm, hz, hiso, horth, happrox⟩ :=
exists_channel_integrals hB hAd hind hcross H hH hc C hC hb T
obtain ⟨Y, hY, hmY, hsY, heY⟩ := PredictableL2.exists_orthogonal_sum I hI2 horth
(by simpa only [hiso] using henergy)
have hsumterm (s : Finset ι) :
AEStronglyMeasurable[F T] (fun ω => ∑ a ∈ s, I a ω) P := by
classical
induction s using Finset.induction_on with
| empty => simpa only [Finset.sum_empty] using
(stronglyMeasurable_const.aestronglyMeasurable :
AEStronglyMeasurable[F T] (fun _ : Ω => (0 : ℝ)) P)
| @insert a s has ih =>
simpa only [Finset.sum_insert has, Pi.add_def] using (hIterm a).add ih
have htermLp : AEStronglyMeasurable[F T] (hY.toLp Y : Ω → ℝ) P := by
apply (isClosed_aestronglyMeasurable (F.le T)).mem_of_tendsto hsY
apply Filter.Eventually.of_forall
intro s
change AEStronglyMeasurable[F T] (↑(∑ a ∈ s, (hI2 a).toLp (I a)) : Ω → ℝ) P
rw [← PredictableL2.toLp_finsetSum I hI2 s]
exact (hsumterm s).congr (MemLp.coeFn_toLp _).symm
have hterminal : AEStronglyMeasurable[F T] Y P := htermLp.congr hY.coeFn_toLp
refine ⟨I, Y, hI2, happrox, hmY, hterminal, hY, ?_, ?_, ?_⟩
· exact PredictableL2.integral_eq_zero_of_hasSum I hI2 hY hsY hz
· simpa only [hiso] using heY
· exact PredictableL2.tendsto_integral_sq_error_of_hasSum I hI2 hY hsY
end Problem367.BrownianArrayFiltration
end
end OAI
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