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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 | import Mathlib.Probability.Martingale.Basic
namespace OAI
noncomputable section
open MeasureTheory Filter Set
namespace Problem367.MartingalePushforward
theorem martingale_of_pullback
{ι Ω Ξ : Type*} [Preorder ι] [mΩ : MeasurableSpace Ω] [mΞ : MeasurableSpace Ξ]
{μ : Measure Ω} {ν : Measure Ξ} [IsFiniteMeasure μ] [IsFiniteMeasure ν]
{φ : Ω → Ξ} (hφ : MeasurePreserving φ μ ν)
{F : Filtration ι mΩ} {G : Filtration ι mΞ}
(hpast : ∀ t, (G t).comap φ ≤ F t)
{X : ι → Ξ → ℝ} (hX : StronglyAdapted G X)
(hM : Martingale (fun t ω => X t (φ ω)) F μ) : Martingale X G ν := by
have hm (t : ι) : AEStronglyMeasurable (X t) ν :=
((hX t).mono (G.le t)).aestronglyMeasurable
have hi (t : ι) : Integrable (X t) ν :=
(hφ.integrable_comp (hm t)).mp (hM.integrable t)
refine ⟨hX, fun s t hst => ?_⟩
apply Filter.EventuallyEq.symm
apply ae_eq_condExp_of_forall_setIntegral_eq (G.le s) (hi t)
(fun A _ _ => (hi s).integrableOn) _ (hX s).aestronglyMeasurable
intro A hA _
have hpre : MeasurableSet[F s] (φ ⁻¹' A) :=
hpast s _ ((comap_measurable (m := G s) φ) hA)
have he (u : ι) : (∫ x in A, X u x ∂ν) = ∫ ω in φ ⁻¹' A, X u (φ ω) ∂μ := by
rw [← hφ.map_eq]
exact setIntegral_map (G.le s _ hA) (by simpa only [hφ.map_eq] using hm u)
hφ.measurable.aemeasurable
rw [he s, he t]
exact hM.setIntegral_eq hst hpre
end Problem367.MartingalePushforward
end
end OAI
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