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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 | import OAI.Probability.FreeIsing.Energy.DiscrepancyEnergy
namespace OAI
noncomputable section
open MeasureTheory Set Filter
open scoped NNReal
namespace Problem367.DiscrepancyEnergy
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
theorem martingale_integral_eq {I : Type*} [Preorder I]
{ℱ : Filtration I mΩ} [SigmaFiniteFiltration μ ℱ]
{M : I → Ω → ℝ} (hM : Martingale M ℱ μ) {s t : I} (hst : s ≤ t) :
(∫ ω, M t ω ∂μ) = ∫ ω, M s ω ∂μ := by
simpa only [Measure.restrict_univ] using
(hM.setIntegral_eq hst (s := Set.univ) MeasurableSet.univ).symm
theorem expectedSquare_ac_and_deriv_of_compensatedSquare_martingale
[SFinite μ] {ℱ : Filtration ℝ≥0 mΩ} [SigmaFiniteFiltration μ ℱ]
{X : ℝ≥0 → Ω → ℝ} {F : ℝ → Ω → ℝ} (T : ℝ≥0)
(hF : Integrable (fun p : ℝ × Ω => F p.1 p.2)
((volume.restrict (Ioc 0 (T : ℝ))).prod μ))
(hM : Martingale
(fun t ω => (X t ω) ^ 2 - ∫ s in (0 : ℝ)..(t : ℝ), F s ω) ℱ μ) :
AbsolutelyContinuousOnInterval
(fun t => ∫ ω, (X t.toNNReal ω) ^ 2 ∂μ) 0 (T : ℝ) ∧
(∀ᵐ t ∂volume.restrict (Icc 0 (T : ℝ)),
deriv (fun t => ∫ ω, (X t.toNNReal ω) ^ 2 ∂μ) t =
∫ ω, F t ω ∂μ) := by
let N : ℝ≥0 → Ω → ℝ :=
fun t ω => (X t ω) ^ 2 - ∫ s in (0 : ℝ)..(t : ℝ), F s ω
have hN : Martingale N ℱ μ := hM
have hN0 : N 0 = fun ω => (X 0 ω) ^ 2 := by
funext ω
simp [N]
let M : ℝ → Ω → ℝ := fun t ω => N t.toNNReal ω - N 0 ω
have hXa : Integrable (fun ω => (X 0 ω) ^ 2) μ := by
simpa only [hN0] using hN.integrable 0
have hMint (t : ℝ) : Integrable (M t) μ :=
(hN.integrable t.toNNReal).sub (hN.integrable 0)
have hmean (t : ℝ) : ∫ ω, M t ω ∂μ = 0 := by
change (∫ ω, N t.toNNReal ω - N 0 ω ∂μ) = 0
have heq : (∫ ω, N t.toNNReal ω ∂μ) = ∫ ω, N 0 ω ∂μ :=
martingale_integral_eq hN (s := 0) (t := t.toNNReal) (by positivity)
rw [integral_sub (hN.integrable _) (hN.integrable _), heq, sub_self]
have hdecomp : ∀ t ∈ Icc 0 (T : ℝ), ∀ᵐ ω ∂μ,
(X t.toNNReal ω) ^ 2 = (X (0 : ℝ).toNNReal ω) ^ 2 + M t ω +
∫ s in (0 : ℝ)..t, F s ω := by
intro t ht
filter_upwards with ω
simp only [M, N, NNReal.coe_zero, Real.toNNReal_zero,
intervalIntegral.integral_same, sub_zero, Real.coe_toNNReal t ht.1]
ring
have hstart : Integrable (fun ω => (X (0 : ℝ).toNNReal ω) ^ 2) μ := by
simpa only [Real.toNNReal_zero] using hXa
exact ⟨secondMoment_absolutelyContinuous T.2 hstart
(fun t _ => hMint t) (fun t _ => hmean t) hF hdecomp,
secondMoment_ae_deriv T.2 hstart
(fun t _ => hMint t) (fun t _ => hmean t) hF hdecomp⟩
theorem secondMoment_nonneg {X : ℝ → Ω → ℝ} (t : ℝ) :
0 ≤ ∫ ω, (X t ω) ^ 2 ∂μ :=
integral_nonneg fun _ => sq_nonneg _
theorem secondMoment_le_of_ae_abs_le [IsProbabilityMeasure μ]
{X : ℝ → Ω → ℝ} {t H : ℝ}
(hInt : Integrable (fun ω => (X t ω) ^ 2) μ)
(hbound : ∀ᵐ ω ∂μ, |X t ω| ≤ H) :
(∫ ω, (X t ω) ^ 2 ∂μ) ≤ H ^ 2 := by
have hsq : ∀ᵐ ω ∂μ, (X t ω) ^ 2 ≤ H ^ 2 := by
filter_upwards [hbound] with ω hω
simpa only [sq_abs] using
(sq_le_sq₀ (abs_nonneg _) ((abs_nonneg _).trans hω)).2 hω
calc
(∫ ω, (X t ω) ^ 2 ∂μ) ≤ ∫ _ : Ω, H ^ 2 ∂μ := integral_mono_ae hInt (integrable_const _) hsq
_ = H ^ 2 := by simp
end Problem367.DiscrepancyEnergy
end
end OAI
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