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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | import OAI.Probability.FreeIsing.Brownian.Martingale
import OAI.Probability.FreeIsing.Martingales.InnovationMartingale
import OAI.Probability.FreeIsing.Trees.PathSplit
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal
namespace Problem367.ConditionalCopyBrownian
variable {O X E : Type*} [MeasurableSpace O] [MeasurableSpace X] [MeasurableSpace E]
def joinedHistory (s : ℝ≥0) (past : O → X)
(q : (ℝ≥0 → E) × (O × O)) : (Set.Iic s → E) × (X × X) :=
(PathSplit.pastPath s q.1, (past q.2.1, past q.2.2))
theorem measurable_joinedHistory (s : ℝ≥0) {past : O → X}
(hpast : Measurable past) : Measurable (joinedHistory (E := E) s past) := by
exact ((PathSplit.measurable_pastPath s).comp measurable_fst).prodMk
((hpast.comp (measurable_fst.comp measurable_snd)).prodMk
(hpast.comp (measurable_snd.comp measurable_snd)))
abbrev joinedPast (s : ℝ≥0) (past : O → X) :
MeasurableSpace ((ℝ≥0 → E) × (O × O)) :=
MeasurableSpace.comap (joinedHistory s past) inferInstance
theorem joinedPast_le (s : ℝ≥0) {past : O → X} (hpast : Measurable past) :
joinedPast s past ≤ (inferInstance : MeasurableSpace ((ℝ≥0 → E) × (O × O))) :=
(measurable_joinedHistory s hpast).comap_le
variable [StandardBorelSpace X] [Nonempty X]
theorem condExp_future_eq_integral
[AddCommGroup E] [MeasurableAdd₂ E] [MeasurableSub₂ E]
(s : ℝ≥0) (rho : Measure (ℝ≥0 → E)) [IsProbabilityMeasure rho]
(eta : Kernel (ℝ≥0 → E) O) [IsMarkovKernel eta]
(past : O → X) (hpast : Measurable past)
(hind : IndepFun
(fun q : (ℝ≥0 → E) × O => (PathSplit.pastPath s q.1, past q.2))
(fun q => PathSplit.futureIncrements s q.1) (rho ⊗ₘ eta))
(f : (Set.Ioi s → E) → ℝ) (hf : Measurable f)
(hfint : Integrable (fun w => f (PathSplit.futureIncrements s w)) rho) :
(rho ⊗ₘ (eta ×ₖ eta))[
(fun q : (ℝ≥0 → E) × (O × O) => f (PathSplit.futureIncrements s q.1)) |
joinedPast s past] =ᵐ[rho ⊗ₘ (eta ×ₖ eta)]
fun _ => ∫ w, f (PathSplit.futureIncrements s w) ∂rho := by
have hi := (PathSplit.indepFun_future_conditionalCopies s rho eta past hpast hind).symm.comp
hf measurable_id
have hfm : Measurable (fun q : (ℝ≥0 → E) × (O × O) =>
f (PathSplit.futureIncrements s q.1)) :=
hf.comp ((PathSplit.measurable_futureIncrements s).comp measurable_fst)
have hce := condExp_indep_eq hfm.comap_le (joinedPast_le s hpast)
(measurable_iff_comap_le.mpr le_rfl).stronglyMeasurable hi
have hmp : MeasurePreserving (Prod.fst : (ℝ≥0 → E) × (O × O) → (ℝ≥0 → E))
(rho ⊗ₘ (eta ×ₖ eta)) rho := ⟨measurable_fst, Measure.fst_compProd _ _⟩
have hint := hmp.integrable_comp_of_integrable hfint
have hmean : (∫ q : (ℝ≥0 → E) × (O × O),
f (PathSplit.futureIncrements s q.1) ∂rho ⊗ₘ (eta ×ₖ eta)) =
∫ w, f (PathSplit.futureIncrements s w) ∂rho := by
simpa only [Function.comp_apply, integral_const, probReal_univ,
one_smul] using (Measure.integral_compProd hint)
simpa only [Function.comp_def, hmean] using hce
theorem condExp_increment_eq_zero
(rho : Measure (ℝ≥0 → ℝ)) [IsProbabilityMeasure rho]
(hB : IsPreBrownianReal (fun t (w : ℝ≥0 → ℝ) => w t) rho)
(eta : Kernel (ℝ≥0 → ℝ) O) [IsMarkovKernel eta]
{s t : ℝ≥0} (hst : s ≤ t)
(past : O → X) (hpast : Measurable past)
(hind : IndepFun
(fun q : (ℝ≥0 → 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 → ℝ) × (O × O) => q.1 t - q.1 s) |
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 → ℝ => w t - w s) 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⟩) (measurable_pi_apply _) hi
have hz : (∫ w : ℝ≥0 → ℝ, w t - w s ∂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, hz] using hc
theorem condExp_increment_sq_eq_elapsed
(rho : Measure (ℝ≥0 → ℝ)) [IsProbabilityMeasure rho]
(hB : IsPreBrownianReal (fun t (w : ℝ≥0 → ℝ) => w t) rho)
(eta : Kernel (ℝ≥0 → ℝ) O) [IsMarkovKernel eta]
{s t : ℝ≥0} (hst : s ≤ t)
(past : O → X) (hpast : Measurable past)
(hind : IndepFun
(fun q : (ℝ≥0 → 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 → ℝ) × (O × O) => (q.1 t - q.1 s) ^ 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 → ℝ => (w t - w s) ^ 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⟩) ^ 2) ((measurable_pi_apply _).pow_const 2) hi
have hz : (∫ w : ℝ≥0 → ℝ, (w t - w s) ^ 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, hz] using hc
theorem isPreBrownian_commonNoise
(rho : Measure (ℝ≥0 → ℝ)) [IsProbabilityMeasure rho]
(hB : IsPreBrownianReal (fun t (w : ℝ≥0 → ℝ) => w t) rho)
(eta : Kernel (ℝ≥0 → ℝ) O) [IsMarkovKernel eta] :
IsPreBrownianReal (fun t (q : (ℝ≥0 → ℝ) × (O × O)) => q.1 t)
(rho ⊗ₘ (eta ×ₖ eta)) := by
have hmp : MeasurePreserving (Prod.fst : (ℝ≥0 → ℝ) × (O × O) → (ℝ≥0 → ℝ))
(rho ⊗ₘ (eta ×ₖ eta)) rho := ⟨measurable_fst, Measure.fst_compProd _ _⟩
exact ⟨fun I => (hB.hasLaw I).comp hmp.hasLaw⟩
theorem commonNoise_martingale
(rho : Measure (ℝ≥0 → ℝ)) [IsProbabilityMeasure rho]
(hB : IsPreBrownianReal (fun t (w : ℝ≥0 → ℝ) => w t) rho)
(eta : Kernel (ℝ≥0 → ℝ) O) [IsMarkovKernel eta]
(past : ℝ≥0 → O → X) (hpast : ∀ s, Measurable (past s))
(G : Filtration ℝ≥0 (inferInstance : MeasurableSpace ((ℝ≥0 → ℝ) × (O × O))))
(hG : ∀ s, G s = joinedPast s (past s))
(hind : ∀ s, IndepFun
(fun q : (ℝ≥0 → 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 → ℝ) × (O × O)) => q.1 t) G
(rho ⊗ₘ (eta ×ₖ eta)) := by
have had : StronglyAdapted G
(fun t (q : (ℝ≥0 → ℝ) × (O × O)) => q.1 t) := by
intro t
rw [hG t]
have hh : Measurable[joinedPast (E := ℝ) t (past t)]
(joinedHistory (E := ℝ) t (past t)) :=
comap_measurable _
exact ((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 rho hB eta).integrable_eval t)
intro s t hst
rw [hG s]
exact condExp_increment_eq_zero rho hB eta hst (past s) (hpast s) (hind s)
end Problem367.ConditionalCopyBrownian
end
end OAI
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