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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 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 | import OAI.Probability.FreeIsing.Gaussian.ContinuousNoiseSplit
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.ContinuousCopyBrownian
open WienerSampler ContinuousObservationHistory ContinuousNoiseSplit
variable {ι : Type*}
abbrev Joint := (ι → Path) × ((ι → Path) × (ι → Path))
def history (s : ℝ≥0) (q : Joint (ι := ι)) : Joint (ι := ι) :=
(stoppedArray s q.1, (stoppedArray s q.2.1, stoppedArray s q.2.2))
theorem measurable_history (s : ℝ≥0) : Measurable (history (ι := ι) s) :=
((measurable_stoppedArray s).comp measurable_fst).prodMk
(((measurable_stoppedArray s).comp (measurable_fst.comp measurable_snd)).prodMk
((measurable_stoppedArray s).comp (measurable_snd.comp measurable_snd)))
theorem history_history {s t : ℝ≥0} (hst : s ≤ t) (q : Joint (ι := ι)) :
history s (history t q) = history s q := by
simp only [history, stoppedArray_stoppedArray hst]
abbrev pastSpace (s : ℝ≥0) : MeasurableSpace (Joint (ι := ι)) :=
MeasurableSpace.comap (history s) inferInstance
def joinedFiltration : Filtration ℝ≥0 (inferInstance : MeasurableSpace (Joint (ι := ι))) where
seq := pastSpace
mono' := by
intro s t hst
have h := MeasurableSpace.comap_mono (g := history (ι := ι) t) (measurable_history s).comap_le
simpa only [MeasurableSpace.comap_comp, Function.comp_def, history_history hst] using h
le' s := (measurable_history s).comap_le
theorem isBrownian_commonNoise
(i : ι) (rho : Measure (ι → Path)) [IsProbabilityMeasure rho]
(hB : IsPreBrownianReal (fun t (w : ι → Path) => w i t) rho)
(eta : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel eta] :
IsBrownianReal (fun t (q : Joint (ι := ι)) => q.1 i t) (rho ⊗ₘ (eta ×ₖ eta)) := by
have hmp : MeasurePreserving (Prod.fst : Joint (ι := ι) → (ι → Path))
(rho ⊗ₘ (eta ×ₖ eta)) rho := ⟨measurable_fst, Measure.fst_compProd _ _⟩
exact ⟨⟨fun I => (hB.hasLaw I).comp hmp.hasLaw⟩,
Filter.Eventually.of_forall (fun q => (q.1 i).continuous)⟩
theorem condExp_increment_eq_zero [Countable ι]
(i : ι) (rho : Measure (ι → Path)) [IsProbabilityMeasure rho]
(hB : IsPreBrownianReal (fun t (w : ι → Path) => w i t) rho)
(eta : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel eta]
{s t : ℝ≥0} (hst : s ≤ t)
(hind : IndepFun (fun q : (ι → Path) × (ι → Path) =>
(stoppedArray s q.1, stoppedArray s q.2))
(fun q => futureArray s q.1) (rho ⊗ₘ eta)) :
(rho ⊗ₘ (eta ×ₖ eta))[(fun q : Joint (ι := ι) => q.1 i t - q.1 i s) |
pastSpace s] =ᵐ[rho ⊗ₘ (eta ×ₖ eta)] fun _ => 0 := by
have hcopy := ContinuousNoiseSplit.indepFun_future_conditionalCopies s rho eta
(stoppedArray s) (measurable_stoppedArray s) hind
have hf : Measurable (fun w : ι → Path => w i (t - s)) :=
(continuous_eval_const (t - s)).measurable.comp (measurable_pi_apply i)
have hi : IndepFun (fun q : Joint (ι := ι) => q.1 i t - q.1 i s) (history s)
(rho ⊗ₘ (eta ×ₖ eta)) := by
change IndepFun (fun q : Joint (ι := ι) => q.1 i t - q.1 i s)
(fun q => (stoppedArray s q.1, (stoppedArray s q.2.1, stoppedArray s q.2.2))) _
simpa only [Function.comp_def, id_eq, futureArray, futurePath_apply,
add_tsub_cancel_of_le hst] using hcopy.symm.comp hf measurable_id
have hm : Measurable (fun q : Joint (ι := ι) => q.1 i t - q.1 i s) :=
(((continuous_eval_const t).measurable.comp (measurable_pi_apply i)).comp measurable_fst).sub
(((continuous_eval_const s).measurable.comp (measurable_pi_apply i)).comp measurable_fst)
have hc := condExp_indep_eq hm.comap_le (measurable_history s).comap_le
(measurable_iff_comap_le.mpr le_rfl).stronglyMeasurable hi
have hBc := (isBrownian_commonNoise i rho hB eta).toIsPreBrownianReal
have hz : (∫ q : Joint (ι := ι), q.1 i t - q.1 i s ∂rho ⊗ₘ (eta ×ₖ eta)) = 0 := by
rw [integral_sub (hBc.integrable_eval t) (hBc.integrable_eval s),
hBc.integral_eval, hBc.integral_eval, sub_self]
simpa only [hz] using hc
theorem commonNoise_martingale [Countable ι]
(i : ι) (rho : Measure (ι → Path)) [IsProbabilityMeasure rho]
(hB : IsPreBrownianReal (fun t (w : ι → Path) => w i t) rho)
(eta : Kernel (ι → Path) (ι → Path)) [IsMarkovKernel eta]
(hind : ∀ s, IndepFun (fun q : (ι → Path) × (ι → Path) =>
(stoppedArray s q.1, stoppedArray s q.2))
(fun q => futureArray s q.1) (rho ⊗ₘ eta)) :
Martingale (fun t (q : Joint (ι := ι)) => q.1 i t) joinedFiltration
(rho ⊗ₘ (eta ×ₖ eta)) := by
have had : StronglyAdapted (joinedFiltration (ι := ι))
(fun t (q : Joint (ι := ι)) => q.1 i t) := by
intro t
change StronglyMeasurable[pastSpace (ι := ι) t] (fun q : Joint (ι := ι) => q.1 i t)
have hh : Measurable[pastSpace (ι := ι) t] (history (ι := ι) t) := comap_measurable _
have hm := (continuous_eval_const t).measurable.comp
((measurable_pi_apply i).comp (measurable_fst.comp hh))
simpa only [Function.comp_def, history, stoppedArray, stoppedPath_apply, min_self] using hm.stronglyMeasurable
apply martingale_of_conditional_increment_zero had
(fun t => (isBrownian_commonNoise i rho hB eta).toIsPreBrownianReal.integrable_eval t)
intro s t hst
exact condExp_increment_eq_zero i rho hB eta hst (hind s)
end Problem367.ContinuousCopyBrownian
end
end OAI
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