| import OAI.Probability.FreeIsing.Conditioning.CopyBrownian |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal |
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| namespace Problem367.ConditionalCopyArrayBrownian |
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| open ConditionalCopyBrownian |
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| variable {ι O X : Type*} [MeasurableSpace O] [MeasurableSpace X] |
| variable [StandardBorelSpace X] [Nonempty X] |
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| theorem condExp_increment_eq_zero |
| (i : ι) (rho : Measure (ℝ≥0 → ι → ℝ)) [IsProbabilityMeasure rho] |
| (hB : IsPreBrownianReal (fun t (w : ℝ≥0 → ι → ℝ) => w t i) rho) |
| (eta : Kernel (ℝ≥0 → ι → ℝ) O) [IsMarkovKernel eta] |
| {s t : ℝ≥0} (hst : s ≤ t) |
| (past : O → X) (hpast : Measurable past) |
| (hind : IndepFun |
| (fun q : (ℝ≥0 → ι → ℝ) × O => (PathSplit.pastPath s q.1, past q.2)) |
| (fun q => PathSplit.futureIncrements s q.1) (rho ⊗ₘ eta)) : |
| (rho ⊗ₘ (eta ×ₖ eta))[ |
| (fun q : (ℝ≥0 → ι → ℝ) × (O × O) => q.1 t i - q.1 s i) | |
| joinedPast s past] =ᵐ[rho ⊗ₘ (eta ×ₖ eta)] fun _ => 0 := by |
| rcases hst.eq_or_lt with rfl | hst |
| · simp |
| have hi : Integrable (fun w : ℝ≥0 → ι → ℝ => w t i - w s i) rho := |
| (hB.integrable_eval t).sub (hB.integrable_eval s) |
| have hc := condExp_future_eq_integral s rho eta past hpast hind |
| (fun z => z ⟨t, hst⟩ i) ((measurable_pi_apply i).comp (measurable_pi_apply _)) hi |
| have hz : (∫ w : ℝ≥0 → ι → ℝ, w t i - w s i ∂rho) = 0 := by |
| rw [integral_sub (hB.integrable_eval t) (hB.integrable_eval s), |
| hB.integral_eval, hB.integral_eval, sub_self] |
| simpa only [PathSplit.futureIncrements, Pi.sub_apply, hz] using hc |
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| theorem condExp_increment_sq_eq_elapsed |
| (i : ι) (rho : Measure (ℝ≥0 → ι → ℝ)) [IsProbabilityMeasure rho] |
| (hB : IsPreBrownianReal (fun t (w : ℝ≥0 → ι → ℝ) => w t i) rho) |
| (eta : Kernel (ℝ≥0 → ι → ℝ) O) [IsMarkovKernel eta] |
| {s t : ℝ≥0} (hst : s ≤ t) |
| (past : O → X) (hpast : Measurable past) |
| (hind : IndepFun |
| (fun q : (ℝ≥0 → ι → ℝ) × O => (PathSplit.pastPath s q.1, past q.2)) |
| (fun q => PathSplit.futureIncrements s q.1) (rho ⊗ₘ eta)) : |
| (rho ⊗ₘ (eta ×ₖ eta))[ |
| (fun q : (ℝ≥0 → ι → ℝ) × (O × O) => (q.1 t i - q.1 s i) ^ 2) | |
| joinedPast s past] =ᵐ[rho ⊗ₘ (eta ×ₖ eta)] |
| fun _ => (t : ℝ) - (s : ℝ) := by |
| rcases hst.eq_or_lt with rfl | hst |
| · simp |
| have hi : Integrable (fun w : ℝ≥0 → ι → ℝ => (w t i - w s i) ^ 2) rho := |
| ((hB.isGaussianProcess.hasGaussianLaw_eval t).memLp_two.sub |
| (hB.isGaussianProcess.hasGaussianLaw_eval s).memLp_two).integrable_sq |
| have hc := condExp_future_eq_integral s rho eta past hpast hind |
| (fun z => (z ⟨t, hst⟩ i) ^ 2) (((measurable_pi_apply i).comp (measurable_pi_apply _)).pow_const 2) hi |
| have hz : (∫ w : ℝ≥0 → ι → ℝ, (w t i - w s i) ^ 2 ∂rho) = (t : ℝ) - (s : ℝ) := by |
| have h := preBrownian_integral_sq (hB.shift s) (t - s) |
| simpa [add_tsub_cancel_of_le hst.le, NNReal.coe_sub hst.le] using h |
| simpa only [PathSplit.futureIncrements, Pi.sub_apply, hz] using hc |
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| theorem isPreBrownian_commonNoise |
| (i : ι) (rho : Measure (ℝ≥0 → ι → ℝ)) [IsProbabilityMeasure rho] |
| (hB : IsPreBrownianReal (fun t (w : ℝ≥0 → ι → ℝ) => w t i) rho) |
| (eta : Kernel (ℝ≥0 → ι → ℝ) O) [IsMarkovKernel eta] : |
| IsPreBrownianReal (fun t (q : (ℝ≥0 → ι → ℝ) × (O × O)) => q.1 t i) |
| (rho ⊗ₘ (eta ×ₖ eta)) := by |
| have hmp : MeasurePreserving (Prod.fst : (ℝ≥0 → ι → ℝ) × (O × O) → (ℝ≥0 → ι → ℝ)) |
| (rho ⊗ₘ (eta ×ₖ eta)) rho := ⟨measurable_fst, Measure.fst_compProd _ _⟩ |
| exact ⟨fun I => (hB.hasLaw I).comp hmp.hasLaw⟩ |
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| theorem commonNoise_martingale |
| (i : ι) (rho : Measure (ℝ≥0 → ι → ℝ)) [IsProbabilityMeasure rho] |
| (hB : IsPreBrownianReal (fun t (w : ℝ≥0 → ι → ℝ) => w t i) rho) |
| (eta : Kernel (ℝ≥0 → ι → ℝ) O) [IsMarkovKernel eta] |
| (past : ℝ≥0 → O → X) (hpast : ∀ s, Measurable (past s)) |
| (G : Filtration ℝ≥0 (inferInstance : MeasurableSpace ((ℝ≥0 → ι → ℝ) × (O × O)))) |
| (hG : ∀ s, G s = joinedPast s (past s)) |
| (hind : ∀ s, IndepFun |
| (fun q : (ℝ≥0 → ι → ℝ) × O => (PathSplit.pastPath s q.1, past s q.2)) |
| (fun q => PathSplit.futureIncrements s q.1) (rho ⊗ₘ eta)) : |
| Martingale (fun t (q : (ℝ≥0 → ι → ℝ) × (O × O)) => q.1 t i) G |
| (rho ⊗ₘ (eta ×ₖ eta)) := by |
| have had : StronglyAdapted G |
| (fun t (q : (ℝ≥0 → ι → ℝ) × (O × O)) => q.1 t i) := by |
| intro t |
| rw [hG t] |
| have hh : Measurable[joinedPast (E := ι → ℝ) t (past t)] |
| (joinedHistory (E := ι → ℝ) t (past t)) := |
| comap_measurable _ |
| exact ((measurable_pi_apply i).comp ((measurable_pi_apply |
| (⟨t, by simp⟩ : Set.Iic t)).comp (measurable_fst.comp hh))).stronglyMeasurable |
| apply martingale_of_conditional_increment_zero had |
| (fun t => (isPreBrownian_commonNoise i rho hB eta).integrable_eval t) |
| intro s t hst |
| rw [hG s] |
| exact condExp_increment_eq_zero i rho hB eta hst (past s) (hpast s) (hind s) |
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| end Problem367.ConditionalCopyArrayBrownian |
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| end |
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| end OAI |
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