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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 | import OAI.Probability.FreeIsing.Brownian.ArrayMoments
import OAI.Probability.FreeIsing.Brownian.Innovation
import OAI.Probability.FreeIsing.Estimates.ProgressiveArrayInnovation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
theorem ae_array_innovation_increment_drift_bound
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
[∀ j, IsProbabilityMeasure (μ j)]
{B : ι → ℝ≥0 → Ω → ℝ} (hmB : ∀ i t, Measurable (B i t))
{σ : S → ι → ℝ} (hσ : Measurable σ) (i : ι)
{C : ℝ≥0} (hσbound : ∀ᵐ s ∂ν, |σ s i| ≤ (C : ℝ))
{m : ℝ≥0 → S × (ι → Ω) → ℝ}
(hm : IsStronglyProgressive (arrayObservationFiltration B hmB σ hσ) m)
(hposterior : ∀ t, m t =ᵐ[ν.prod (Measure.infinitePi μ)]
(ν.prod (Measure.infinitePi μ))[(fun q : S × (ι → Ω) => σ q.1 i) |
arrayObservationFiltration B hmB σ hσ t])
{s t : ℝ≥0} (hst : s ≤ t) :
∀ᵐ q ∂ν.prod (Measure.infinitePi μ),
|((plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q) -
(plantedArrayObservation B σ i s q - nnrealTimePrimitive m s q)) -
(B i t (q.2 i) - B i s (q.2 i))| ≤
(2 * (C : ℝ)) * ((t : ℝ) - (s : ℝ)) := by
have hsignal : ∀ᵐ q ∂ν.prod (Measure.infinitePi μ), |σ q.1 i| ≤ (C : ℝ) :=
Measure.quasiMeasurePreserving_fst.ae hσbound
filter_upwards [hsignal,
ae_lipschitzWith_array_posterior_primitive ν μ hmB hσ i hσbound hm hposterior]
with q hsq hlip
have htime : 0 ≤ (t : ℝ) - (s : ℝ) := sub_nonneg.mpr hst
have hprim : |nnrealTimePrimitive m t q - nnrealTimePrimitive m s q| ≤
(C : ℝ) * ((t : ℝ) - (s : ℝ)) := by
simpa only [Real.dist_eq, NNReal.dist_eq, abs_of_nonneg htime] using
hlip.dist_le_mul t s
have heq : ((plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q) -
(plantedArrayObservation B σ i s q - nnrealTimePrimitive m s q)) -
(B i t (q.2 i) - B i s (q.2 i)) =
((t : ℝ) - (s : ℝ)) * σ q.1 i -
(nnrealTimePrimitive m t q - nnrealTimePrimitive m s q) := by
unfold plantedArrayObservation
ring
rw [heq]
calc
_ ≤ |((t : ℝ) - (s : ℝ)) * σ q.1 i| +
|nnrealTimePrimitive m t q - nnrealTimePrimitive m s q| := abs_sub _ _
_ ≤ ((t : ℝ) - (s : ℝ)) * (C : ℝ) +
(C : ℝ) * ((t : ℝ) - (s : ℝ)) := by
rw [abs_mul, abs_of_nonneg htime]
exact add_le_add (mul_le_mul_of_nonneg_left hsq htime) hprim
_ = _ := by ring
theorem preBrownian_array_innovation_of_progressive_posterior
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
{B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ i, IsPreBrownianReal (B i) (μ i))
(hmB : ∀ i t, Measurable (B i t))
{σ : S → ι → ℝ} (hσ : Measurable σ) (i : ι)
{C : ℝ≥0} (hσbound : ∀ᵐ s ∂ν, |σ s i| ≤ (C : ℝ))
{m : ℝ≥0 → S × (ι → Ω) → ℝ}
(hm : IsStronglyProgressive (arrayObservationFiltration B hmB σ hσ) m)
(hposterior : ∀ t, m t =ᵐ[ν.prod (Measure.infinitePi μ)]
(ν.prod (Measure.infinitePi μ))[(fun q : S × (ι → Ω) => σ q.1 i) |
arrayObservationFiltration B hmB σ hσ t]) :
IsPreBrownianReal
(fun t q => plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q)
(ν.prod (Measure.infinitePi μ)) := by
classical
have (j : ι) : IsProbabilityMeasure (μ j) :=
(hB j).isGaussianProcess.isProbabilityMeasure
apply BrownianInnovation.isPreBrownianReal_of_bounded_drift
(planted_coordinate_preBrownian ν μ hB i)
(preBrownian_array_innovation_martingale_of_progressive_posterior
ν μ hB hmB hσ i hσbound hm hposterior)
(L := 2 * (C : ℝ))
· intro s t hst
have hv := planted_weighted_increment_sq_condExp_subfield ν μ hB hmB hst
({i} : Finset ι) (fun _ => (1 : ℝ))
(arrayObservationFiltration_le_signalArrayFiltration B hmB σ hσ s)
simpa using hv
· positivity
· intro s t hst
exact ae_array_innovation_increment_drift_bound ν μ hmB hσ i hσbound hm hposterior hst
· have hzero := (planted_noise_coordinate_measurePreserving ν μ i).quasiMeasurePreserving.ae
(hB i).eval_zero_ae_eq_zero
filter_upwards [hzero] with q hq
simpa only [plantedArrayObservation, nnrealTimePrimitive, NNReal.coe_zero,
zero_mul, add_zero, intervalIntegral.integral_same, sub_zero] using hq
theorem brownian_array_innovation_of_progressive_posterior
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
{B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ i, IsBrownianReal (B i) (μ i))
(hmB : ∀ i t, Measurable (B i t))
{σ : S → ι → ℝ} (hσ : Measurable σ) (i : ι)
{C : ℝ≥0} (hσbound : ∀ᵐ s ∂ν, |σ s i| ≤ (C : ℝ))
{m : ℝ≥0 → S × (ι → Ω) → ℝ}
(hm : IsStronglyProgressive (arrayObservationFiltration B hmB σ hσ) m)
(hposterior : ∀ t, m t =ᵐ[ν.prod (Measure.infinitePi μ)]
(ν.prod (Measure.infinitePi μ))[(fun q : S × (ι → Ω) => σ q.1 i) |
arrayObservationFiltration B hmB σ hσ t]) :
IsBrownianReal
(fun t q => plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q)
(ν.prod (Measure.infinitePi μ)) :=
⟨preBrownian_array_innovation_of_progressive_posterior
ν μ (fun j => (hB j).toIsPreBrownianReal) hmB hσ i hσbound hm hposterior,
(brownian_array_innovation_regular_martingale ν μ hB hmB hσ i hσbound hm hposterior).2.1⟩
end Problem367
end
end OAI
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