openAI-math-dataset / lean /OAI /Probability /FreeIsing /Martingales /ContinuousMartingaleVariation.lean
| import OAI.Probability.FreeIsing.Martingales.BoundedMartingaleVariation | |
| import OAI.Probability.FreeIsing.Martingales.QuadraticVariation | |
| import OAI.Probability.FreeIsing.Stochastic.CompensatorVariation | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology BigOperators | |
| namespace Problem367.ContinuousMartingaleVariation | |
| open MartingaleQuadraticVariation | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} | |
| theorem tendsto_integral_weighted_error_sq [IsProbabilityMeasure μ] | |
| {I : Type*} [Preorder I] [PseudoMetricSpace 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) | |
| {C : ℝ} (hC : 0 ≤ C) (hMb : ∀ t, ∀ᵐ ω ∂μ, |M t ω| ≤ C) | |
| {S : Set I} (hS : IsCompact S) | |
| (hMc : ∀ᵐ ω ∂μ, ContinuousOn (fun t => M t ω) S) | |
| (hAc : ∀ᵐ ω ∂μ, ContinuousOn (fun t => A t ω) S) | |
| (t : ℕ → ℕ → I) (ht : ∀ n, Monotone (t n)) (N : ℕ → ℕ) | |
| (hmem : ∀ n i, i ≤ N n → t n i ∈ S) (δ : ℕ → ℝ) | |
| (hδ : Tendsto δ atTop (𝓝 0)) | |
| (hmesh : ∀ n i, i < N n → dist (t n (i + 1)) (t n i) ≤ δ n) | |
| (hApos : ∀ᵐ ω ∂μ, ∀ n i, i < N n → | |
| 0 ≤ A (t n (i + 1)) ω - A (t n i) ω) | |
| {B : ℝ} (hB : 0 ≤ B) | |
| (hAb : ∀ᵐ ω ∂μ, ∀ n, A (t n (N n)) ω - A (t n 0) ω ≤ B) | |
| (H : ℕ → ℕ → Ω → ℝ) | |
| (hH : ∀ n i, i < N n → StronglyMeasurable[F (t n i)] (H n i)) | |
| {L : ℝ} (hL : 0 ≤ L) | |
| (hHb : ∀ n i, i < N n → ∀ᵐ ω ∂μ, |H n i ω| ≤ L) : | |
| Tendsto (fun n => ∫ ω, (∑ i ∈ Finset.range (N n), | |
| H n i ω * squareError M A (t n i) (t n (i + 1)) ω) ^ 2 ∂μ) | |
| atTop (𝓝 0) := by | |
| have hMm (s : I) : AEStronglyMeasurable (M s) μ := | |
| ((hM.stronglyMeasurable s).mono (F.le s)).aestronglyMeasurable | |
| have hM2 (s : I) : MemLp (M s) 2 μ := | |
| MemLp.of_bound (hMm s) C (by simpa only [Real.norm_eq_abs] using hMb s) | |
| have hAm (s : I) : AEStronglyMeasurable (A s) μ := | |
| ((stronglyAdapted_compensator hM hQ s).mono (F.le s)).aestronglyMeasurable | |
| have hD4 (n i : ℕ) (_hi : i < N n) : | |
| MemLp (fun ω => (increment M (t n i) (t n (i + 1)) ω) ^ 2) 2 μ := by | |
| apply BoundedMartingaleVariation.bounded_pow_memLp_two | |
| ((hMm (t n (i + 1))).sub (hMm (t n i))) (show 0 ≤ 2 * C by positivity) ?_ 2 | |
| filter_upwards [hMb (t n (i + 1)), hMb (t n i)] with ω hn hi | |
| exact (abs_sub _ _).trans (by linarith) | |
| have hA2 (n i : ℕ) (hi : i < N n) : | |
| MemLp (increment A (t n i) (t n (i + 1))) 2 μ := | |
| CompensatorVariation.memLp_increment_of_total_bound (fun j => hAm (t n j)) | |
| (hApos.mono (fun ω hω => hω n)) (hAb.mono (fun ω hω => hω n)) hi 2 | |
| have hfour := BoundedMartingaleVariation.tendsto_sum_integral_fourth_of_continuous | |
| hM hC hMb hS hMc t ht N hmem δ hδ hmesh | |
| have hcomp := CompensatorVariation.tendsto_sum_integral_sq_increments | |
| hS hAm hAc t N δ hmem | |
| (fun n i hi => by simpa only [dist_comm] using hmesh n i hi) hδ hApos hB hAb | |
| exact tendsto_integral_finite_grid_weighted_error_sq hM hQ hM2 t ht N H hH | |
| hD4 hA2 hL hHb hfour hcomp | |
| theorem tendsto_integral_weighted_error_sq_of_monotone [IsProbabilityMeasure μ] | |
| {I : Type*} [Preorder I] [PseudoMetricSpace 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) | |
| {C : ℝ} (hC : 0 ≤ C) (hMb : ∀ t, ∀ᵐ ω ∂μ, |M t ω| ≤ C) | |
| {S : Set I} (hS : IsCompact S) | |
| (hMc : ∀ᵐ ω ∂μ, ContinuousOn (fun t => M t ω) S) | |
| (hAc : ∀ᵐ ω ∂μ, ContinuousOn (fun t => A t ω) S) | |
| (hAmono : ∀ᵐ ω ∂μ, MonotoneOn (fun t => A t ω) S) | |
| (t : ℕ → ℕ → I) (ht : ∀ n, Monotone (t n)) (N : ℕ → ℕ) | |
| (hmem : ∀ n i, i ≤ N n → t n i ∈ S) (δ : ℕ → ℝ) | |
| (hδ : Tendsto δ atTop (𝓝 0)) | |
| (hmesh : ∀ n i, i < N n → dist (t n (i + 1)) (t n i) ≤ δ n) | |
| {B : ℝ} (hB : 0 ≤ B) | |
| (hAb : ∀ᵐ ω ∂μ, ∀ n, A (t n (N n)) ω - A (t n 0) ω ≤ B) | |
| (H : ℕ → ℕ → Ω → ℝ) | |
| (hH : ∀ n i, i < N n → StronglyMeasurable[F (t n i)] (H n i)) | |
| {L : ℝ} (hL : 0 ≤ L) | |
| (hHb : ∀ n i, i < N n → ∀ᵐ ω ∂μ, |H n i ω| ≤ L) : | |
| Tendsto (fun n => ∫ ω, (∑ i ∈ Finset.range (N n), | |
| H n i ω * squareError M A (t n i) (t n (i + 1)) ω) ^ 2 ∂μ) | |
| atTop (𝓝 0) := by | |
| apply tendsto_integral_weighted_error_sq hM hQ hC hMb hS hMc hAc t ht N hmem δ hδ | |
| hmesh ?_ hB hAb H hH hL hHb | |
| filter_upwards [hAmono] with ω hω | |
| intro n i hi | |
| exact sub_nonneg.mpr (hω (hmem n i hi.le) (hmem n (i + 1) (Nat.succ_le_of_lt hi)) | |
| (ht n (Nat.le_succ i))) | |
| end Problem367.ContinuousMartingaleVariation | |
| end | |
| end OAI | |