| import OAI.Probability.FreeIsing.Stochastic.SimpleIntegralIsometry |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology |
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| namespace Problem367.MartingaleTransformL2 |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} |
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| 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 |
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| 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)) |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 _)) |
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| 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)) |
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| 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) |
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| end Problem367.MartingaleTransformL2 |
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| end |
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| end OAI |
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