| import Mathlib.Probability.Martingale.Basic |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter Set |
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| namespace Problem367.MartingalePushforward |
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| 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 |
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| end Problem367.MartingalePushforward |
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| end |
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| end OAI |
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