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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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | import OAI.Probability.FreeIsing.Conditioning.CopyBrownian
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal
namespace Problem367.ConditionalCopyArrayBrownian
open ConditionalCopyBrownian
variable {ι O X : Type*} [MeasurableSpace O] [MeasurableSpace X]
variable [StandardBorelSpace X] [Nonempty X]
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
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
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⟩
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)
end Problem367.ConditionalCopyArrayBrownian
end
end OAI
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