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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 | import OAI.Probability.FreeIsing.Stochastic.SimpleIntegralIsometry
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Problem367.MartingaleTransformL2
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
theorem condExp_increment_zero [IsFiniteMeasure μ]
{ι : Type*} [Preorder ι] {F : Filtration ι mΩ} {M : ι → Ω → ℝ}
(hM : Martingale M F μ) {s t : ι} (hst : s ≤ t) :
μ[(fun ω => M t ω - M s ω) | F s] =ᵐ[μ] fun _ => 0 := by
change μ[M t - M s | F s] =ᵐ[μ] fun _ => 0
have hsub := condExp_sub (hM.integrable t) (hM.integrable s) (F s)
have hs := hM.condExp_ae_eq (le_refl s)
filter_upwards [hsub, hM.condExp_ae_eq hst, hs] with ω hω htω hsω
simpa only [Pi.sub_apply, htω, hsω, sub_self] using hω
theorem bounded_mul_memLp_two {H D : Ω → ℝ} {K : ℝ}
(hH : AEStronglyMeasurable H μ) (hD : MemLp D 2 μ)
(hbound : ∀ᵐ ω ∂μ, |H ω| ≤ K) :
MemLp (fun ω => H ω * D ω) 2 μ := by
apply hD.of_le_mul (c := K) (hH.mul hD.aestronglyMeasurable)
filter_upwards [hbound] with ω hω
simpa only [Real.norm_eq_abs, Pi.mul_apply, abs_mul] using
mul_le_mul_of_nonneg_right hω (abs_nonneg (D ω))
theorem integral_sq_transform [IsFiniteMeasure μ]
{F : Filtration ℕ mΩ} {M H : ℕ → Ω → ℝ}
(hM : Martingale M F μ) (hM2 : ∀ i, MemLp (M i) 2 μ)
(hH : StronglyAdapted F H) {K : ℝ}
(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 * (M (i + 1) ω - M i ω)) ^ 2 ∂μ := by
let D : ℕ → Ω → ℝ := fun i ω => M (i + 1) ω - M i ω
have hD (i : ℕ) : StronglyMeasurable[F (i + 1)] (D i) :=
(hM.stronglyMeasurable (i + 1)).sub
((hM.stronglyMeasurable i).mono (F.mono (Nat.le_succ i)))
have hD2 (i : ℕ) : MemLp (D i) 2 μ := (hM2 (i + 1)).sub (hM2 i)
have hHD (i : ℕ) : MemLp (fun ω => H i ω * D i ω) 2 μ :=
bounded_mul_memLp_two ((hH i).mono (F.le i)).aestronglyMeasurable
(hD2 i) (hbound i)
have hzero (i : ℕ) : μ[D i | F i] =ᵐ[μ] 0 :=
condExp_increment_zero hM (Nat.le_succ i)
apply SimpleIntegral.integral_sq_sum_of_orthogonal s (fun i ω => H i ω * D i ω)
· intro i hi j hj
exact (hHD i).integrable_mul (hHD j)
· intro i hi j hj hij
rcases lt_or_gt_of_ne hij with hij | hji
· exact SimpleIntegral.predictable_increments_orthogonal F H D hH hD
(fun i => (hD2 i).integrable (by norm_num)) hHD hzero hij
· simpa only [mul_comm] using
SimpleIntegral.predictable_increments_orthogonal F H D hH hD
(fun i => (hD2 i).integrable (by norm_num)) hHD hzero hji
theorem sum_integral_sq_increment [IsFiniteMeasure μ]
{F : Filtration ℕ mΩ} {M : ℕ → Ω → ℝ}
(hM : Martingale M F μ) (hM2 : ∀ i, MemLp (M i) 2 μ) (n : ℕ) :
(∑ i ∈ Finset.range n, ∫ ω, (M (i + 1) ω - M i ω) ^ 2 ∂μ) =
∫ ω, (M n ω - M 0 ω) ^ 2 ∂μ := by
have h := integral_sq_transform hM hM2
(H := fun _ _ => (1 : ℝ)) (fun _ => stronglyMeasurable_const)
(K := 1) (fun _ => Filter.Eventually.of_forall (fun _ => by simp))
(Finset.range n)
simp only [one_mul] at h
rw [← h]
congr 1
funext ω
congr 1
clear h
induction n with
| zero => simp
| succ n ih =>
rw [Finset.sum_range_succ, ih]
ring
theorem integral_sq_transform_le [IsFiniteMeasure μ]
{F : Filtration ℕ mΩ} {M H : ℕ → Ω → ℝ}
(hM : Martingale M F μ) (hM2 : ∀ i, MemLp (M i) 2 μ)
(hH : StronglyAdapted F H) {K : ℝ} (hK : 0 ≤ K)
(hbound : ∀ i, ∀ᵐ ω ∂μ, |H i ω| ≤ K) (n : ℕ) :
(∫ ω, (∑ i ∈ Finset.range n,
H i ω * (M (i + 1) ω - M i ω)) ^ 2 ∂μ) ≤
K ^ 2 * ∫ ω, (M n ω - M 0 ω) ^ 2 ∂μ := by
rw [integral_sq_transform hM hM2 hH hbound,
← sum_integral_sq_increment hM hM2, Finset.mul_sum]
apply Finset.sum_le_sum
intro i hi
have hD2 := (hM2 (i + 1)).sub (hM2 i)
have hHD := bounded_mul_memLp_two
((hH i).mono (F.le i)).aestronglyMeasurable hD2 (hbound i)
rw [← integral_const_mul]
apply integral_mono_ae hHD.integrable_sq (hD2.integrable_sq.const_mul (K ^ 2))
filter_upwards [hbound i] with ω hω
rw [mul_pow]
apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)
simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hK).2 hω
theorem integral_sq_increment_eq [IsFiniteMeasure μ]
{ι : Type*} [Preorder ι] {F : Filtration ι mΩ} {M : ι → Ω → ℝ}
(hM : Martingale M F μ) {s t : ι} (hst : s ≤ t)
(hs2 : MemLp (M s) 2 μ) (ht2 : MemLp (M t) 2 μ) :
(∫ ω, (M t ω - M s ω) ^ 2 ∂μ) =
(∫ ω, M t ω ^ 2 ∂μ) - ∫ ω, M s ω ^ 2 ∂μ := by
have hD2 : MemLp (fun ω => M t ω - M s ω) 2 μ := ht2.sub hs2
have hcross : Integrable (fun ω => M s ω * (M t ω - M s ω)) μ :=
hs2.integrable_mul hD2
have horth : (∫ ω, M s ω * (M t ω - M s ω) ∂μ) = 0 :=
SimpleIntegral.integral_mul_eq_zero_of_condExp_eq_zero (F.le s)
(hM.stronglyMeasurable s) (hD2.integrable (by norm_num)) hcross
(condExp_increment_zero hM hst)
have heq : (∫ ω, M t ω ^ 2 ∂μ) =
(∫ ω, (M t ω - M s ω) ^ 2 ∂μ) +
2 * (∫ ω, M s ω * (M t ω - M s ω) ∂μ) +
∫ ω, M s ω ^ 2 ∂μ := by
calc
_ = ∫ ω, (M t ω - M s ω) ^ 2 +
2 * (M s ω * (M t ω - M s ω)) + M s ω ^ 2 ∂μ := by
apply integral_congr_ae
exact Filter.Eventually.of_forall (fun ω => by ring)
_ = _ := by
have h1 := integral_add hD2.integrable_sq (hcross.const_mul 2)
have h2 := integral_add (hD2.integrable_sq.add (hcross.const_mul 2)) hs2.integrable_sq
simp only [Pi.add_apply] at h1 h2
rw [h1, integral_const_mul] at h2
exact h2
rw [horth] at heq
linarith
theorem integral_sq_increment_le [IsFiniteMeasure μ]
{ι : Type*} [Preorder ι] {F : Filtration ι mΩ} {M : ι → Ω → ℝ}
(hM : Martingale M F μ) {s t : ι} (hst : s ≤ t)
(hs2 : MemLp (M s) 2 μ) (ht2 : MemLp (M t) 2 μ) :
(∫ ω, (M t ω - M s ω) ^ 2 ∂μ) ≤ ∫ ω, M t ω ^ 2 ∂μ := by
rw [integral_sq_increment_eq hM hst hs2 ht2]
exact sub_le_self _ (integral_nonneg (fun ω => sq_nonneg _))
theorem integral_sq_grid_transform_le_terminal [IsFiniteMeasure μ]
{ι : Type*} [Preorder ι] {F : Filtration ι mΩ} {M : ι → Ω → ℝ}
(hM : Martingale M F μ) (hM2 : ∀ t, MemLp (M t) 2 μ)
(t : ℕ → ι) (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) :
(∫ ω, (∑ i ∈ Finset.range n,
H i ω * (M (t (i + 1)) ω - M (t i) ω)) ^ 2 ∂μ) ≤
K ^ 2 * ∫ ω, M (t n) ω ^ 2 ∂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 :=
⟨by intro i; exact hM.stronglyMeasurable (t i),
by intro i j hij; exact hM.condExp_ae_eq (ht hij)⟩
exact (integral_sq_transform_le hMG (fun i => hM2 (t i)) hH hK hbound n).trans
(mul_le_mul_of_nonneg_left
(integral_sq_increment_le hM (ht (Nat.zero_le n)) (hM2 _) (hM2 _)) (sq_nonneg K))
theorem tendsto_integral_sq_grid_transform [IsFiniteMeasure μ]
{ι : Type*} [Preorder ι] {F : Filtration ι mΩ}
{M : ℕ → ι → Ω → ℝ} (hM : ∀ k, Martingale (M k) F μ)
(hM2 : ∀ k t, MemLp (M k t) 2 μ) (t : ℕ → ℕ → ι)
(ht : ∀ k, Monotone (t k)) (N : ℕ → ℕ) {T : ι}
(hterminal : ∀ k, t k (N k) = T) (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)
(hlim : Tendsto (fun k => ∫ ω, M k T ω ^ 2 ∂μ) atTop (𝓝 0)) :
Tendsto (fun k => ∫ ω, (∑ i ∈ Finset.range (N k),
H k i ω * (M k (t k (i + 1)) ω - M k (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 (K ^ 2))
simpa only [hterminal k] using
integral_sq_grid_transform_le_terminal (hM k) (hM2 k) (t k) (ht k)
(H k) (hH k) hK (hbound k) (N k)
end Problem367.MartingaleTransformL2
end
end OAI
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