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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Probability.FreeIsing.Brownian.FiltrationIntegral
import OAI.Probability.FreeIsing.Brownian.GridRefinement
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.BrownianGrid
theorem integralSum_succ_sub {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
(t : ℕ → ℝ≥0) (H : ℕ → Ω → ℝ) (n : ℕ) (ω : Ω) :
integralSum B t H (n + 1) ω - integralSum B t H n ω =
H n ω * (B (t (n + 1)) ω - B (t n) ω) := by
simp only [integralSum, Finset.sum_range_succ, add_sub_cancel_left]
theorem integralSum_compose {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
(t : ℕ → ℝ≥0) (H K : ℕ → Ω → ℝ) (n : ℕ) (ω : Ω) :
(∑ i ∈ Finset.range n, K i ω *
(integralSum B t H (i + 1) ω - integralSum B t H i ω)) =
integralSum B t (fun i ω => K i ω * H i ω) n ω := by
simp_rw [integralSum_succ_sub]
simp only [integralSum, mul_assoc]
theorem integralSum_compose_refined {Ω : Type*} (B : ℝ≥0 → Ω → ℝ)
(t : ℕ → ℝ≥0) (H K : ℕ → Ω → ℝ) (n : ℕ) {k : ℕ}
(hk : 0 < k) (ω : Ω) :
(∑ j ∈ Finset.range n, K j ω *
(integralSum B t H ((j + 1) * k) ω - integralSum B t H (j * k) ω)) =
integralSum B t (fun i ω => K (i / k) ω * H i ω) (n * k) ω := by
rw [weighted_sum_refinement (fun i => integralSum B t H i ω)
(fun j => K j ω) n hk]
simp_rw [integralSum_succ_sub]
simp only [integralSum, mul_assoc]
theorem stronglyMeasurable_compose_refined
{Ω : Type*} [mΩ : MeasurableSpace Ω] (F : Filtration ℝ≥0 mΩ)
{t : ℕ → ℝ≥0} (ht : Monotone t) {H K : ℕ → Ω → ℝ} {k : ℕ}
(hH : ∀ i, StronglyMeasurable[F (t i)] (H i))
(hK : ∀ j, StronglyMeasurable[F (t (j * k))] (K j)) (i : ℕ) :
StronglyMeasurable[F (t i)] (fun ω => K (i / k) ω * H i ω) :=
((hK (i / k)).mono (F.mono (ht (Nat.div_mul_le_self i k)))).mul (hH i)
theorem memLp_compose_of_bounded
{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω}
{H K : Ω → ℝ} (hH : MemLp H 2 P) (hK : AEStronglyMeasurable K P)
{C : ℝ} (hb : ∀ᵐ ω ∂P, |K ω| ≤ C) :
MemLp (fun ω => K ω * H ω) 2 P := by
apply hH.of_le_mul (c := C) (hK.mul hH.aestronglyMeasurable)
filter_upwards [hb] with ω hω
simpa only [Real.norm_eq_abs, Pi.mul_apply, abs_mul] using
mul_le_mul_of_nonneg_right hω (abs_nonneg (H ω))
theorem composed_integralSum_difference_isometry
{Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
{B : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ}
(hB : IsPreBrownianReal B P) (hAd : StronglyAdapted F B)
(hind : ∀ s t, s ≤ t → Indep
(MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P)
(t : ℕ → ℝ≥0) (ht : Monotone t) (H K J : ℕ → Ω → ℝ)
{k : ℕ} (hk : 0 < k)
(hH : ∀ i, StronglyMeasurable[F (t i)] (H i))
(hK : ∀ j, StronglyMeasurable[F (t (j * k))] (K j))
(hJ : ∀ i, StronglyMeasurable[F (t i)] (J i))
(hKH2 : ∀ i, MemLp (fun ω => K (i / k) ω * H i ω) 2 P)
(hJ2 : ∀ i, MemLp (J i) 2 P) (n : ℕ) :
(∫ ω, ((∑ j ∈ Finset.range n, K j ω *
(integralSum B t H ((j + 1) * k) ω - integralSum B t H (j * k) ω)) -
integralSum B t J (n * k) ω) ^ 2 ∂P) =
∑ i ∈ Finset.range (n * k), ((t (i + 1) : ℝ) - (t i : ℝ)) *
∫ ω, (K (i / k) ω * H i ω - J i ω) ^ 2 ∂P := by
simp_rw [integralSum_compose_refined B t H K n hk]
simpa only [integralSum, sub_mul, Finset.sum_sub_distrib] using
BrownianFiltration.predictable_sum_isometry hB hAd hind t ht
(fun i ω => K (i / k) ω * H i ω - J i ω)
(fun i => (stronglyMeasurable_compose_refined F ht hH hK i).sub (hJ i))
(fun i => (hKH2 i).sub (hJ2 i)) (Finset.range (n * k))
theorem composed_integralSum_difference_bound
{Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
{B : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ}
(hB : IsPreBrownianReal B P) (hAd : StronglyAdapted F B)
(hind : ∀ s t, s ≤ t → Indep
(MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P)
(t : ℕ → ℝ≥0) (ht : Monotone t) (H K J : ℕ → Ω → ℝ)
{k : ℕ} (hk : 0 < k)
(hH : ∀ i, StronglyMeasurable[F (t i)] (H i))
(hK : ∀ j, StronglyMeasurable[F (t (j * k))] (K j))
(hJ : ∀ i, StronglyMeasurable[F (t i)] (J i))
(hKH2 : ∀ i, MemLp (fun ω => K (i / k) ω * H i ω) 2 P)
(hJ2 : ∀ i, MemLp (J i) 2 P) (n : ℕ) {ε : ℝ}
(happrox : ∀ i < n * k,
(∫ ω, (K (i / k) ω * H i ω - J i ω) ^ 2 ∂P) ≤ ε) :
(∫ ω, ((∑ j ∈ Finset.range n, K j ω *
(integralSum B t H ((j + 1) * k) ω - integralSum B t H (j * k) ω)) -
integralSum B t J (n * k) ω) ^ 2 ∂P) ≤
((t (n * k) : ℝ) - (t 0 : ℝ)) * ε := by
rw [composed_integralSum_difference_isometry hB hAd hind t ht H K J hk
hH hK hJ hKH2 hJ2 n]
calc
_ ≤ ∑ i ∈ Finset.range (n * k), ((t (i + 1) : ℝ) - (t i : ℝ)) * ε := by
apply Finset.sum_le_sum
intro i hi
exact mul_le_mul_of_nonneg_left (happrox i (Finset.mem_range.mp hi))
(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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