| import OAI.Probability.FreeIsing.Brownian.SimpleIntegral |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367.BrownianGrid |
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| def integralSum {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℝ≥0) |
| (H : ℕ → Ω → ℝ) (n : ℕ) (ω : Ω) : ℝ := |
| ∑ i ∈ Finset.range n, H i ω * (B (t (i + 1)) ω - B (t i) ω) |
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| lemma sum_range_blocks {E : Type*} [AddCommMonoid E] (f : ℕ → E) (n k : ℕ) : |
| ∑ i ∈ Finset.range (n * k), f i = |
| ∑ j ∈ Finset.range n, ∑ l ∈ Finset.range k, f (j * k + l) := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| rw [Nat.succ_mul, Finset.sum_range_add, ih, Finset.sum_range_succ] |
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| lemma weighted_sum_refinement (x H : ℕ → ℝ) (n : ℕ) {k : ℕ} (hk : 0 < k) : |
| (∑ j ∈ Finset.range n, H j * (x ((j + 1) * k) - x (j * k))) = |
| ∑ i ∈ Finset.range (n * k), H (i / k) * (x (i + 1) - x i) := by |
| rw [sum_range_blocks] |
| apply Finset.sum_congr rfl |
| intro j hj |
| calc |
| H j * (x ((j + 1) * k) - x (j * k)) = |
| H j * (∑ l ∈ Finset.range k, (x (j * k + (l + 1)) - x (j * k + l))) := by |
| rw [Finset.sum_range_sub (fun l => x (j * k + l)) k] |
| simp [Nat.add_mul] |
| _ = ∑ l ∈ Finset.range k, H j * (x (j * k + (l + 1)) - x (j * k + l)) := |
| Finset.mul_sum _ _ _ |
| _ = _ := by |
| apply Finset.sum_congr rfl |
| intro l hl |
| have hdiv : (j * k + l) / k = j := by |
| rw [Nat.mul_comm j k, Nat.mul_add_div hk, Nat.div_eq_of_lt (Finset.mem_range.mp hl)] |
| omega |
| rw [hdiv] |
| congr 3 |
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| theorem integralSum_refinement {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) |
| (t : ℕ → ℝ≥0) (H : ℕ → Ω → ℝ) (n : ℕ) {k : ℕ} (hk : 0 < k) : |
| integralSum B (fun j => t (j * k)) H n = |
| integralSum B t (fun i => H (i / k)) (n * k) := by |
| funext ω |
| exact weighted_sum_refinement (fun i => B (t i) ω) (fun i => H i ω) n hk |
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| lemma integralSum_sub {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (t : ℕ → ℝ≥0) |
| (H K : ℕ → Ω → ℝ) (n : ℕ) : |
| integralSum B t (fun i ω => H i ω - K i ω) n = |
| fun ω => integralSum B t H n ω - integralSum B t K n ω := by |
| funext ω |
| simp only [integralSum, sub_mul, Finset.sum_sub_distrib] |
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| theorem stronglyMeasurable_refined {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {B : ℝ≥0 → Ω → ℝ} (hm : ∀ t, StronglyMeasurable (B t)) |
| {t : ℕ → ℝ≥0} (ht : Monotone t) {H : ℕ → Ω → ℝ} {k : ℕ} |
| (hH : ∀ j, StronglyMeasurable[Filtration.natural B hm (t (j * k))] (H j)) |
| (i : ℕ) : StronglyMeasurable[Filtration.natural B hm (t i)] (H (i / k)) := |
| (hH (i / k)).mono ((Filtration.natural B hm).mono (ht (Nat.div_mul_le_self i k))) |
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| theorem preBrownian_difference_isometry |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hm : ∀ t, StronglyMeasurable (B t)) |
| (t : ℕ → ℝ≥0) (ht : Monotone t) (H K : ℕ → Ω → ℝ) |
| {k : ℕ} (hk : 0 < k) |
| (hH : ∀ j, StronglyMeasurable[Filtration.natural B hm (t (j * k))] (H j)) |
| (hK : ∀ i, StronglyMeasurable[Filtration.natural B hm (t i)] (K i)) |
| (hH2 : ∀ j, MemLp (H j) 2 P) (hK2 : ∀ i, MemLp (K i) 2 P) (n : ℕ) : |
| (∫ ω, (integralSum B (fun j => t (j * k)) H n ω - |
| integralSum B t K (n * k) ω) ^ 2 ∂P) = |
| ∑ i ∈ Finset.range (n * k), ((t (i + 1) : ℝ) - (t i : ℝ)) * |
| ∫ ω, (H (i / k) ω - K i ω) ^ 2 ∂P := by |
| rw [integralSum_refinement B t H n hk] |
| simpa only [integralSum, sub_mul, Finset.sum_sub_distrib] using |
| preBrownian_predictable_sum_isometry hB hm t ht |
| (fun i ω => H (i / k) ω - K i ω) |
| (fun i => (stronglyMeasurable_refined hm ht hH i).sub (hK i)) |
| (fun i => (hH2 (i / k)).sub (hK2 i)) (Finset.range (n * k)) |
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| theorem preBrownian_difference_bound |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hm : ∀ t, StronglyMeasurable (B t)) |
| (t : ℕ → ℝ≥0) (ht : Monotone t) (H K : ℕ → Ω → ℝ) |
| {k : ℕ} (hk : 0 < k) |
| (hH : ∀ j, StronglyMeasurable[Filtration.natural B hm (t (j * k))] (H j)) |
| (hK : ∀ i, StronglyMeasurable[Filtration.natural B hm (t i)] (K i)) |
| (hH2 : ∀ j, MemLp (H j) 2 P) (hK2 : ∀ i, MemLp (K i) 2 P) |
| (n : ℕ) {ε : ℝ} |
| (happrox : ∀ i < n * k, (∫ ω, (H (i / k) ω - K i ω) ^ 2 ∂P) ≤ ε) : |
| (∫ ω, (integralSum B (fun j => t (j * k)) H n ω - |
| integralSum B t K (n * k) ω) ^ 2 ∂P) ≤ |
| ((t (n * k) : ℝ) - (t 0 : ℝ)) * ε := by |
| rw [preBrownian_difference_isometry hB hm t ht H K hk hH hK hH2 hK2 n] |
| calc |
| _ ≤ ∑ i ∈ Finset.range (n * k), ((t (i + 1) : ℝ) - (t i : ℝ)) * ε := by |
| apply Finset.sum_le_sum |
| intro i hi |
| apply mul_le_mul_of_nonneg_left (happrox i (Finset.mem_range.mp hi)) |
| exact sub_nonneg.mpr (NNReal.coe_le_coe.mpr (ht (Nat.le_succ i))) |
| _ = _ := by rw [← Finset.sum_mul, Finset.sum_range_sub (fun i => (t i : ℝ)) (n * k)] |
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| end Problem367.BrownianGrid |
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| end |
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| end OAI |
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