| 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 | |