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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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Filter
namespace Problem367
theorem integral_of_conditional_versions
{T Ω : Type*} [mT : MeasurableSpace T] (m : MeasurableSpace Ω)
[mΩ : MeasurableSpace Ω] (ν : Measure T) (μ : Measure Ω) [SFinite ν] [SFinite μ]
(hm : m ≤ mΩ) [SigmaFinite (μ.trim hm)]
{F G : T × Ω → ℝ}
(hF : Integrable F (ν.prod μ)) (hG : Integrable G (ν.prod μ))
(hGm : StronglyMeasurable[@MeasurableSpace.prod T Ω mT m] G)
(hcond : ∀ᵐ t ∂ν,
(fun ω => G (t, ω)) =ᵐ[μ] μ[(fun ω => F (t, ω)) | m]) :
(fun ω => ∫ t, G (t, ω) ∂ν) =ᵐ[μ]
μ[(fun ω => ∫ t, F (t, ω) ∂ν) | m] := by
apply ae_eq_condExp_of_forall_setIntegral_eq hm hF.integral_prod_right
· intro s hs hfin
exact hG.integral_prod_right.integrableOn
· intro s hs hfin
have hFrestrict : Integrable F (ν.prod (μ.restrict s)) :=
hF.mono_measure (Measure.prod_mono le_rfl Measure.restrict_le_self)
have hGrestrict : Integrable G (ν.prod (μ.restrict s)) :=
hG.mono_measure (Measure.prod_mono le_rfl Measure.restrict_le_self)
calc
(∫ ω in s, ∫ t, G (t, ω) ∂ν ∂μ) =
∫ t, ∫ ω in s, G (t, ω) ∂μ ∂ν :=
(integral_integral_swap (f := fun t ω => G (t, ω)) hGrestrict).symm
_ = ∫ t, ∫ ω in s, F (t, ω) ∂μ ∂ν := by
apply integral_congr_ae
filter_upwards [hcond, hF.prod_right_ae] with t ht hFt
exact (setIntegral_congr_ae (hm s hs) (ht.mono (fun ω hω _ => hω))).trans
(setIntegral_condExp hm hFt hs)
_ = ∫ ω in s, ∫ t, F (t, ω) ∂ν ∂μ :=
integral_integral_swap (f := fun t ω => F (t, ω)) hFrestrict
· have hgm : StronglyMeasurable[m] (fun ω => ∫ t, G (t, ω) ∂ν) := by
let : MeasurableSpace Ω := m
exact hGm.integral_prod_left'
exact hgm.aestronglyMeasurable
theorem condExp_setIntegral_martingale
{Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(ℱ : Filtration ℝ mΩ) {f : ℝ → Ω → ℝ} (hf : Martingale f ℱ μ)
{s t : ℝ} (hst : s ≤ t)
(hInt : Integrable (fun p : ℝ × Ω => f p.1 p.2)
((volume.restrict (Set.Ioc s t)).prod μ)) :
μ[(fun ω => ∫ u in Set.Ioc s t, f u ω) | ℱ s] =ᵐ[μ]
(fun ω => (t - s) * f s ω) := by
let ν : Measure ℝ := volume.restrict (Set.Ioc s t)
have hGm : StronglyMeasurable[@MeasurableSpace.prod ℝ Ω inferInstance (ℱ s)]
(fun p : ℝ × Ω => f s p.2) := by
let : MeasurableSpace Ω := ℱ s
exact (hf.stronglyMeasurable s).comp_measurable measurable_snd
have hcond : ∀ᵐ u ∂ν,
(fun ω => f s ω) =ᵐ[μ] μ[(fun ω => f u ω) | ℱ s] := by
filter_upwards [ae_restrict_mem measurableSet_Ioc] with u hu
exact (hf.condExp_ae_eq hu.1.le).symm
have h := integral_of_conditional_versions (ℱ s) ν μ (ℱ.le s)
hInt ((hf.integrable s).comp_snd ν) hGm hcond
simpa [ν, Real.volume_real_Ioc_of_le hst] using h.symm
theorem condExp_intervalIntegral_martingale
{Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(ℱ : Filtration ℝ mΩ) {f : ℝ → Ω → ℝ} (hf : Martingale f ℱ μ)
{s t : ℝ} (hst : s ≤ t)
(hInt : Integrable (fun p : ℝ × Ω => f p.1 p.2)
((volume.restrict (Set.Ioc s t)).prod μ)) :
μ[(fun ω => ∫ u in s..t, f u ω) | ℱ s] =ᵐ[μ]
(fun ω => (t - s) * f s ω) := by
simpa only [intervalIntegral.integral_of_le hst] using
condExp_setIntegral_martingale μ ℱ hf hst hInt
end Problem367
end
end OAI
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