| import OAI.Probability.FreeIsing.Brownian.ContinuousAssociativity | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped NNReal Topology | |
| namespace Problem367.MartingaleIntegralStability | |
| open BrownianGrid BrownianContinuousAssociativity | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} | |
| [IsFiniteMeasure P] {F : Filtration ℝ≥0 mΩ} | |
| {M N K : ℝ≥0 → Ω → ℝ} | |
| theorem uniformSum_difference_sq_le_terminal | |
| (hM : Martingale M F P) (hN : Martingale N F P) | |
| (hM2 : ∀ t, MemLp (M t) 2 P) (hN2 : ∀ t, MemLp (N t) 2 P) | |
| (hK : StronglyAdapted F K) {D : ℝ} (hD : 0 ≤ D) | |
| (hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D) (T : ℝ≥0) (n : ℕ) : | |
| (∫ ω, (uniformSum M K T n ω - uniformSum N K T n ω) ^ 2 ∂P) ≤ | |
| D ^ 2 * ∫ ω, (M T ω - N T ω) ^ 2 ∂P := by | |
| by_cases hn : n = 0 | |
| · subst n | |
| simp only [uniformSum, integralSum, Finset.range_zero, Finset.sum_empty, | |
| sub_self, zero_pow (by decide : 2 ≠ 0), integral_zero] | |
| positivity | |
| have h := MartingaleTransformL2.integral_sq_grid_transform_le_terminal | |
| (hM.sub hN) (fun t => (hM2 t).sub (hN2 t)) | |
| (uniformGrid T n) (uniformGrid_mono T n) (fun j => K (uniformGrid T n j)) | |
| (fun j => hK _) hD (fun j => hbK.mono (fun ω hω => hω _)) n | |
| rw [uniformGrid_end T (Nat.pos_of_ne_zero hn)] at h | |
| simp only [Pi.sub_apply] at h | |
| convert h using 1 | |
| congr 1 | |
| funext ω | |
| simp only [uniformSum, integralSum] | |
| rw [← Finset.sum_sub_distrib] | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| ring | |
| theorem uniformSum_limit_of_integrator_limits | |
| {α : Type*} {l : Filter α} [l.NeBot] | |
| {A : α → ℝ≥0 → Ω → ℝ} {Zs : α → Ω → ℝ} {Z : Ω → ℝ} | |
| (hM : Martingale M F P) (hA : ∀ k, Martingale (A k) F P) | |
| (hM2 : ∀ t, MemLp (M t) 2 P) (hA2 : ∀ k t, MemLp (A k t) 2 P) | |
| (hK : StronglyAdapted F K) {D : ℝ} (hD : 0 ≤ D) | |
| (hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D) | |
| (hZs : ∀ k, MemLp (Zs k) 2 P) (hZ : MemLp Z 2 P) (T : ℝ≥0) | |
| (hterminal : Tendsto (fun k => ∫ ω, (A k T ω - M T ω) ^ 2 ∂P) l (𝓝 0)) | |
| (hendpoints : Tendsto (fun k => ∫ ω, (Zs k ω - Z ω) ^ 2 ∂P) l (𝓝 0)) | |
| (hinner : ∀ k, Tendsto (fun n => ∫ ω, | |
| (uniformSum (A k) K T n ω - Zs k ω) ^ 2 ∂P) atTop (𝓝 0)) : | |
| Tendsto (fun n => ∫ ω, (uniformSum M K T n ω - Z ω) ^ 2 ∂P) | |
| atTop (𝓝 0) := by | |
| apply Metric.tendsto_atTop.mpr | |
| intro ε hε | |
| let η := ε / (16 * (D ^ 2 + 1)) | |
| have hη : 0 < η := div_pos hε (by positivity) | |
| obtain ⟨k, hkM, hkZ⟩ := ((hterminal.eventually (gt_mem_nhds hη)).and | |
| (hendpoints.eventually (gt_mem_nhds hη))).exists | |
| obtain ⟨L, hL⟩ := ((hinner k).eventually (gt_mem_nhds hη)).exists_forall_of_atTop | |
| refine ⟨L, ?_⟩ | |
| intro n hn | |
| have hnerr := hL n hn | |
| have houterM := uniformSum_memLp hM2 hK hbK T n | |
| have houterA := uniformSum_memLp (hA2 k) hK hbK T n | |
| have htriangle := integral_sq_sub_le_two houterM hZ houterA | |
| have htriangle' := integral_sq_sub_le_two hZ houterA (hZs k) | |
| have hcontract := uniformSum_difference_sq_le_terminal hM (hA k) hM2 (hA2 k) | |
| hK hD hbK T n | |
| have hsymM : (∫ ω, (M T ω - A k T ω) ^ 2 ∂P) = | |
| ∫ ω, (A k T ω - M T ω) ^ 2 ∂P := by | |
| apply integral_congr_ae | |
| exact Eventually.of_forall (fun ω => sub_sq_comm _ _) | |
| have hsymZ : (∫ ω, (Z ω - Zs k ω) ^ 2 ∂P) = | |
| ∫ ω, (Zs k ω - Z ω) ^ 2 ∂P := by | |
| apply integral_congr_ae | |
| exact Eventually.of_forall (fun ω => sub_sq_comm _ _) | |
| rw [hsymM] at hcontract | |
| rw [hsymZ] at htriangle' | |
| rw [Real.dist_eq, sub_zero, abs_of_nonneg (integral_nonneg (fun ω => sq_nonneg _))] | |
| have hbound : (∫ ω, (uniformSum M K T n ω - Z ω) ^ 2 ∂P) < | |
| (2 * D ^ 2 + 8) * η := by | |
| have hmBound := mul_le_mul_of_nonneg_left hkM.le (sq_nonneg D) | |
| nlinarith | |
| apply hbound.trans | |
| dsimp only [η] | |
| rw [← mul_div_assoc] | |
| apply (div_lt_iff₀ (by positivity : 0 < 16 * (D ^ 2 + 1))).mpr | |
| nlinarith [sq_nonneg D] | |
| end Problem367.MartingaleIntegralStability | |
| end | |
| end OAI | |