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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 | import OAI.Probability.FreeIsing.Martingales.QuadraticVariation
import OAI.Probability.FreeIsing.Martingales.TransformL2
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators
namespace Problem367.MartingaleBracketIsometry
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
theorem integrable_sq_mul {H D : Ω → ℝ} {K : ℝ} (hK : 0 ≤ K)
(hH : AEStronglyMeasurable H μ) (hD : Integrable D μ)
(hbound : ∀ᵐ ω ∂μ, |H ω| ≤ K) :
Integrable (fun ω => H ω ^ 2 * D ω) μ := by
apply hD.bdd_mul (c := K ^ 2) (hH.pow 2)
filter_upwards [hbound] with ω hω
simpa only [Real.norm_eq_abs, Pi.pow_apply, abs_sq, sq_abs] using
(sq_le_sq₀ (abs_nonneg (H ω)) hK).2 hω
theorem integral_sq_mul_increment [IsFiniteMeasure μ]
{I : Type*} [Preorder I] {F : Filtration I mΩ}
{M A : I → Ω → ℝ} (hM : Martingale M F μ)
(hQ : Martingale (fun t ω => M t ω ^ 2 - A t ω) F μ)
(hM2 : ∀ t, MemLp (M t) 2 μ) {s t : I} (hst : s ≤ t)
{H : Ω → ℝ} (hH : StronglyMeasurable[F s] H)
{K : ℝ} (hK : 0 ≤ K) (hbound : ∀ᵐ ω ∂μ, |H ω| ≤ K) :
(∫ ω, (H ω * (M t ω - M s ω)) ^ 2 ∂μ) =
∫ ω, H ω ^ 2 * (A t ω - A s ω) ∂μ := by
have hHae : AEStronglyMeasurable H μ := (hH.mono (F.le s)).aestronglyMeasurable
have hD2 := (hM2 t).sub (hM2 s)
have hHD2 : MemLp (fun ω => H ω * (M t ω - M s ω)) 2 μ :=
MartingaleTransformL2.bounded_mul_memLp_two hHae hD2 hbound
have hAint : Integrable (fun ω => A t ω - A s ω) μ :=
(MartingaleQuadraticVariation.integrable_compensator hQ hM2 t).sub
(MartingaleQuadraticVariation.integrable_compensator hQ hM2 s)
have hHAint := integrable_sq_mul hK hHae hAint hbound
have herr := MartingaleQuadraticVariation.integrable_squareError hQ hM2 s t
have hHEint := integrable_sq_mul hK hHae herr hbound
have hzero := SimpleIntegral.integral_mul_eq_zero_of_condExp_eq_zero
(F.le s) (hH.pow 2) herr hHEint
(MartingaleQuadraticVariation.condExp_squareError_eq_zero hM hQ hM2 hst)
have hident : (fun ω => H ω ^ 2 *
MartingaleQuadraticVariation.squareError M A s t ω) =
(fun ω => (H ω * (M t ω - M s ω)) ^ 2) -
(fun ω => H ω ^ 2 * (A t ω - A s ω)) := by
funext ω
simp only [MartingaleQuadraticVariation.squareError,
MartingaleQuadraticVariation.increment, Pi.sub_apply]
ring
simp only [Pi.pow_apply] at hzero
rw [hident] at hzero
change (∫ ω, (H ω * (M t ω - M s ω)) ^ 2 -
H ω ^ 2 * (A t ω - A s ω) ∂μ) = 0 at hzero
rw [integral_sub hHD2.integrable_sq hHAint] at hzero
exact sub_eq_zero.mp hzero
theorem integral_sq_transform [IsFiniteMeasure μ]
{F : Filtration ℕ mΩ} {M A H : ℕ → Ω → ℝ}
(hM : Martingale M F μ)
(hQ : Martingale (fun t ω => M t ω ^ 2 - A t ω) F μ)
(hM2 : ∀ t, MemLp (M t) 2 μ) (hH : StronglyAdapted F H)
{K : ℝ} (hK : 0 ≤ K) (hbound : ∀ i, ∀ᵐ ω ∂μ, |H i ω| ≤ K)
(s : Finset ℕ) :
(∫ ω, (∑ i ∈ s, H i ω * (M (i + 1) ω - M i ω)) ^ 2 ∂μ) =
∑ i ∈ s, ∫ ω, H i ω ^ 2 * (A (i + 1) ω - A i ω) ∂μ := by
rw [MartingaleTransformL2.integral_sq_transform hM hM2 hH hbound]
apply Finset.sum_congr rfl
intro i _
exact integral_sq_mul_increment hM hQ hM2 (Nat.le_succ i) (hH i) hK (hbound i)
theorem integral_sq_transform_le_random_envelope [IsFiniteMeasure μ]
{F : Filtration ℕ mΩ} {M A H : ℕ → Ω → ℝ}
(hM : Martingale M F μ)
(hQ : Martingale (fun t ω => M t ω ^ 2 - A t ω) F μ)
(hM2 : ∀ t, MemLp (M t) 2 μ) (hH : StronglyAdapted F H)
{K : ℝ} (hK : 0 ≤ K) (hbound : ∀ i, ∀ᵐ ω ∂μ, |H i ω| ≤ K)
(s : Finset ℕ) {R : Ω → ℝ} (hR : Integrable R μ)
(hRnonneg : ∀ᵐ ω ∂μ, 0 ≤ R ω)
(hEnvelope : ∀ᵐ ω ∂μ, ∀ i ∈ s, H i ω ^ 2 ≤ R ω)
(hA : ∀ᵐ ω ∂μ, ∀ i ∈ s, 0 ≤ A (i + 1) ω - A i ω)
{C : ℝ} (htotal : ∀ᵐ ω ∂μ, (∑ i ∈ s, (A (i + 1) ω - A i ω)) ≤ C) :
(∫ ω, (∑ i ∈ s, H i ω * (M (i + 1) ω - M i ω)) ^ 2 ∂μ) ≤
C * ∫ ω, R ω ∂μ := by
have hint (i : ℕ) : Integrable (fun ω => H i ω ^ 2 *
(A (i + 1) ω - A i ω)) μ :=
integrable_sq_mul hK ((hH i).mono (F.le i)).aestronglyMeasurable
((MartingaleQuadraticVariation.integrable_compensator hQ hM2 (i + 1)).sub
(MartingaleQuadraticVariation.integrable_compensator hQ hM2 i)) (hbound i)
rw [integral_sq_transform hM hQ hM2 hH hK hbound,
← integral_finsetSum s (fun i _ => hint i), ← integral_const_mul]
apply integral_mono_ae (integrable_finsetSum s (fun i _ => hint i)) (hR.const_mul C)
filter_upwards [hRnonneg, hEnvelope, hA, htotal] with ω hRω hEω hAω htω
calc
(∑ i ∈ s, H i ω ^ 2 * (A (i + 1) ω - A i ω)) ≤
∑ i ∈ s, R ω * (A (i + 1) ω - A i ω) := by
exact Finset.sum_le_sum (fun i hi =>
mul_le_mul_of_nonneg_right (hEω i hi) (hAω i hi))
_ = R ω * (∑ i ∈ s, (A (i + 1) ω - A i ω)) := by rw [Finset.mul_sum]
_ ≤ R ω * C := mul_le_mul_of_nonneg_left htω hRω
_ = C * R ω := mul_comm _ _
theorem integral_sq_grid_transform [IsFiniteMeasure μ]
{I : Type*} [Preorder I] {F : Filtration I mΩ} {M A : I → Ω → ℝ}
(hM : Martingale M F μ)
(hQ : Martingale (fun t ω => M t ω ^ 2 - A t ω) F μ)
(hM2 : ∀ t, MemLp (M t) 2 μ)
(t : ℕ → I) (ht : Monotone t) (H : ℕ → Ω → ℝ)
(hH : ∀ i, StronglyMeasurable[F (t i)] (H i))
{K : ℝ} (hK : 0 ≤ K) (hbound : ∀ i, ∀ᵐ ω ∂μ, |H i ω| ≤ K)
(s : Finset ℕ) :
(∫ ω, (∑ i ∈ s, H i ω * (M (t (i + 1)) ω - M (t i) ω)) ^ 2 ∂μ) =
∑ i ∈ s, ∫ ω, H i ω ^ 2 * (A (t (i + 1)) ω - A (t i) ω) ∂μ := by
let G : Filtration ℕ mΩ :=
⟨fun i => F (t i), fun _ _ hij => F.mono (ht hij), fun i => F.le (t i)⟩
have hMG : Martingale (fun i => M (t i)) G μ :=
⟨fun i => hM.stronglyMeasurable (t i), fun _ _ hij => hM.condExp_ae_eq (ht hij)⟩
have hQG : Martingale (fun i ω => M (t i) ω ^ 2 - A (t i) ω) G μ :=
⟨fun i => hQ.stronglyMeasurable (t i), fun _ _ hij => hQ.condExp_ae_eq (ht hij)⟩
exact integral_sq_transform hMG hQG (fun i => hM2 (t i)) hH hK hbound s
theorem integral_sq_grid_transform_le_random_envelope [IsFiniteMeasure μ]
{I : Type*} [Preorder I] {F : Filtration I mΩ} {M A : I → Ω → ℝ}
(hM : Martingale M F μ)
(hQ : Martingale (fun t ω => M t ω ^ 2 - A t ω) F μ)
(hM2 : ∀ t, MemLp (M t) 2 μ)
(t : ℕ → I) (ht : Monotone t) (H : ℕ → Ω → ℝ)
(hH : ∀ i, StronglyMeasurable[F (t i)] (H i))
{K : ℝ} (hK : 0 ≤ K) (hbound : ∀ i, ∀ᵐ ω ∂μ, |H i ω| ≤ K)
(n : ℕ) {R : Ω → ℝ} (hR : Integrable R μ)
(hRnonneg : ∀ᵐ ω ∂μ, 0 ≤ R ω)
(hEnvelope : ∀ᵐ ω ∂μ, ∀ i < n, H i ω ^ 2 ≤ R ω)
(hA : ∀ᵐ ω ∂μ, Monotone (fun u => A u ω))
{C : ℝ} (htotal : ∀ᵐ ω ∂μ, A (t n) ω - A (t 0) ω ≤ C) :
(∫ ω, (∑ i ∈ Finset.range n,
H i ω * (M (t (i + 1)) ω - M (t i) ω)) ^ 2 ∂μ) ≤
C * ∫ ω, R ω ∂μ := by
let G : Filtration ℕ mΩ :=
⟨fun i => F (t i), fun _ _ hij => F.mono (ht hij), fun i => F.le (t i)⟩
have hMG : Martingale (fun i => M (t i)) G μ :=
⟨fun i => hM.stronglyMeasurable (t i), fun _ _ hij => hM.condExp_ae_eq (ht hij)⟩
have hQG : Martingale (fun i ω => M (t i) ω ^ 2 - A (t i) ω) G μ :=
⟨fun i => hQ.stronglyMeasurable (t i), fun _ _ hij => hQ.condExp_ae_eq (ht hij)⟩
apply integral_sq_transform_le_random_envelope hMG hQG
(fun i => hM2 (t i)) hH hK hbound (Finset.range n) hR hRnonneg
· filter_upwards [hEnvelope] with ω hω
exact fun i hi => hω i (Finset.mem_range.mp hi)
· filter_upwards [hA] with ω hω
exact fun i _ => sub_nonneg.mpr (hω (ht (Nat.le_succ i)))
· filter_upwards [htotal] with ω hω
have htel : (∑ i ∈ Finset.range n, (A (t (i + 1)) ω - A (t i) ω)) =
A (t n) ω - A (t 0) ω := by
clear hEnvelope htotal hω
induction n with
| zero => simp
| succ n ih =>
rw [Finset.sum_range_succ, ih]
ring
simpa only [htel] using hω
theorem tendsto_integral_sq_grid_transform [IsFiniteMeasure μ]
{I : Type*} [Preorder I] {F : Filtration I mΩ} {M A : I → Ω → ℝ}
(hM : Martingale M F μ)
(hQ : Martingale (fun t ω => M t ω ^ 2 - A t ω) F μ)
(hM2 : ∀ t, MemLp (M t) 2 μ)
(t : ℕ → ℕ → I) (ht : ∀ k, Monotone (t k))
(H : ℕ → ℕ → Ω → ℝ) (hH : ∀ k i, StronglyMeasurable[F (t k i)] (H k i))
{K : ℝ} (hK : 0 ≤ K) (hbound : ∀ k i, ∀ᵐ ω ∂μ, |H k i ω| ≤ K)
(N : ℕ → ℕ) {R : ℕ → Ω → ℝ} (hR : ∀ k, Integrable (R k) μ)
(hRnonneg : ∀ k, ∀ᵐ ω ∂μ, 0 ≤ R k ω)
(hEnvelope : ∀ k, ∀ᵐ ω ∂μ, ∀ i < N k, H k i ω ^ 2 ≤ R k ω)
(hA : ∀ᵐ ω ∂μ, Monotone (fun u => A u ω))
{C : ℝ} (htotal : ∀ k, ∀ᵐ ω ∂μ, A (t k (N k)) ω - A (t k 0) ω ≤ C)
(hlim : Tendsto (fun k => ∫ ω, R k ω ∂μ) atTop (𝓝 0)) :
Tendsto (fun k => ∫ ω, (∑ i ∈ Finset.range (N k),
H k i ω * (M (t k (i + 1)) ω - M (t k i) ω)) ^ 2 ∂μ) atTop (𝓝 0) := by
apply squeeze_zero (fun k => integral_nonneg (fun ω => sq_nonneg _))
(fun k => ?_) (by simpa using hlim.const_mul C)
exact integral_sq_grid_transform_le_random_envelope hM hQ hM2 (t k) (ht k)
(H k) (hH k) hK (hbound k) (N k) (hR k) (hRnonneg k)
(hEnvelope k) hA (htotal k)
end Problem367.MartingaleBracketIsometry
end
end OAI
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