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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.Probability.FreeIsing.Brownian.SimpleIntegral
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.BrownianGrid
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)
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]
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
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
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]
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)))
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))
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)]
end Problem367.BrownianGrid
end
end OAI
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