File size: 3,403 Bytes
135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Scalar.ScalarTiltedExpectation
namespace OAI
section
noncomputable section
open scoped Topology NNReal ContDiff
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem boundedDerivs_of_derivatives {f g dg : ℝ → ℝ}
(hg : ∀ x, HasDerivAt f (g x) x) (hdg : ∀ x, HasDerivAt g (dg x) x)
(hcont : Continuous dg) {B C : ℝ≥0}
(hB : ∀ x, |g x| ≤ B) (hC : ∀ x, |dg x| ≤ C) : BoundedDerivs f := by
have he : deriv f=g := funext (fun x => (hg x).deriv)
have he' : deriv g=dg := funext (fun x => (hdg x).deriv)
have hgc : ContDiff ℝ 1 g := contDiff_one_iff_deriv.mpr
⟨fun x => (hdg x).differentiableAt,he' ▸ hcont⟩
have hfc : ContDiff ℝ 2 f := by
rw [show (2:ℕ∞ω)=1+1 by norm_num,contDiff_succ_iff_deriv]
exact ⟨fun x => (hg x).differentiableAt,by norm_num,he ▸ hgc⟩
have hfe : fderiv ℝ f=fun x => g x • (ContinuousLinearMap.id ℝ ℝ) := by
funext x
apply ContinuousLinearMap.ext
intro y
simp only [smul_apply,ContinuousLinearMap.id_apply,smul_eq_mul,
fderiv_eq_deriv_mul,he]
refine ⟨hfc,B,C,B.coe_nonneg,C.coe_nonneg,?_,?_⟩
· intro x
rw [← norm_deriv_eq_norm_fderiv,he,Real.norm_eq_abs]
exact hB x
· intro x
rw [hfe,← norm_deriv_eq_norm_fderiv,((hdg x).smul_const (ContinuousLinearMap.id ℝ ℝ)).deriv,
norm_smul,ContinuousLinearMap.norm_id,Real.norm_eq_abs,mul_one]
exact hC x
def boundedPrimitive (g : ℝ → ℝ) (x : ℝ) : ℝ := ∫ y in 0..x, g y
theorem boundedPrimitive_hasDerivAt {g : ℝ → ℝ} (hg : Continuous g) (x : ℝ) :
HasDerivAt (boundedPrimitive g) (g x) x :=
intervalIntegral.integral_hasDerivAt_right (hg.intervalIntegrable 0 x)
hg.stronglyMeasurable.stronglyMeasurableAtFilter hg.continuousAt
theorem boundedPrimitive_regular {g dg : ℝ → ℝ}
(hg : ∀ x, HasDerivAt g (dg x) x) (hdg : Continuous dg)
{B C : ℝ≥0} (hB : ∀ x, |g x| ≤ B) (hC : ∀ x, |dg x| ≤ C) :
BoundedDerivs (boundedPrimitive g) :=
boundedDerivs_of_derivatives
(boundedPrimitive_hasDerivAt (continuous_iff_continuousAt.mpr (fun x => (hg x).continuousAt))) hg hdg hB hC
theorem boundedPrimitive_lipschitz {g : ℝ → ℝ} (hg : Continuous g) {B : ℝ≥0}
(hB : ∀ x, |g x| ≤ B) : LipschitzWith B (boundedPrimitive g) := by
apply lipschitzWith_of_nnnorm_deriv_le (fun x => (boundedPrimitive_hasDerivAt hg x).differentiableAt)
intro x
rw [(boundedPrimitive_hasDerivAt hg x).deriv]
exact_mod_cast hB x
theorem dyadicScalarAverage_tendsto_C1 (β : ℝ) {α : ℝ → ℝ}
(hα : ∀ z, α z ∈ Icc (0:ℝ) 1) (hmono : Monotone α)
{f g dg : ℝ → ℝ} (hf : BoundedDerivs f)
(hg : ∀ y, HasDerivAt g (dg y) y) (hdg : Continuous dg)
{K B C : ℝ≥0} (hfK : LipschitzWith K f)
(hB : ∀ y, |g y| ≤ B) (hC : ∀ y, |dg y| ≤ C)
(s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x : ℝ) :
Tendsto (fun n => dyadicScalarAverage β α n s t f g x) atTop
(𝓝 (scalarCDFAverage β α s t f g x)) := by
have hc : Continuous g := continuous_iff_continuousAt.mpr (fun y => (hg y).continuousAt)
exact dyadicScalarAverage_tendsto β hα hmono hf (boundedPrimitive_regular hg hdg hB hC)
(boundedPrimitive_hasDerivAt hc) hg hfK (boundedPrimitive_lipschitz hc hB) hB hC s t ht x
end SK.Analytic
end
end
end OAI
|