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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | import OAI.Probability.FreeIsing.Martingales.L2MartingaleRightContinuation
import OAI.Probability.FreeIsing.Martingales.SpatialRandomEvaluation
import OAI.Probability.FreeIsing.Stochastic.StoppedIntegralLocality
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.StoppedRowNormalization
open BrownianGrid BrownianIntegralProcess DominatedMartingaleRightContinuation
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
theorem stopped_channel_normalization
{F : Filtration ℝ≥0 mΩ} {B C R K I J : ℝ≥0 → Ω → ℝ}
(hB : IsPreBrownianReal B P) (hAd : StronglyAdapted F B)
(hind : ∀ s t, s ≤ t → Indep
(MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P)
(hcB : ∀ᵐ ω ∂P, Continuous (fun t => B t ω))
(hC : StronglyAdapted F C) (hR : StronglyAdapted F R) (hK : StronglyAdapted F K)
(hcC : ∀ᵐ ω ∂P, Continuous (fun t => C t ω))
(hcK : ∀ᵐ ω ∂P, Continuous (fun t => K t ω))
{a b c : ℝ} (ha : 0 ≤ a) (hc : 0 ≤ c)
(hbC : ∀ᵐ ω ∂P, ∀ t, |C t ω| ≤ a)
(hbR : ∀ᵐ ω ∂P, ∀ t, |R t ω| ≤ b)
(hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ c)
(hI : Martingale I F P) (hJ : Martingale J F P)
(hI2 : ∀ t, MemLp (I t) 2 P) (hJ2 : ∀ t, MemLp (J t) 2 P)
(hcI : ∀ᵐ ω ∂P, Continuous (fun t => I t ω))
(hcJ : ∀ᵐ ω ∂P, Continuous (fun t => J t ω))
(T : ℝ≥0)
(hIlim : Tendsto (fun n => ∫ ω,
(uniformSum B C T n ω - I T ω) ^ 2 ∂P) atTop (𝓝 0))
(hJlim : Tendsto (fun n => ∫ ω,
(uniformSum B R T n ω - J T ω) ^ 2 ∂P) atTop (𝓝 0))
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F.rightCont τ)
(hlocal : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, (t : WithTop ℝ≥0) ≤ τ ω →
K t ω * C t ω = R t ω) :
Tendsto (fun n => ∫ ω,
(uniformSum (stoppedProcess I τ) K T n ω - stoppedProcess J τ T ω) ^ 2 ∂P)
atTop (𝓝 0) := by
let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
have hKC : StronglyAdapted F (fun t ω => K t ω * C t ω) :=
fun t => (hK t).mul (hC t)
have hcKC : ∀ᵐ ω ∂P, Continuous (fun t => K t ω * C t ω) :=
(hcK.and hcC).mono (fun ω hω => hω.1.mul hω.2)
have hbKC : ∀ᵐ ω ∂P, ∀ t, |K t ω * C t ω| ≤ c * a := by
filter_upwards [hbK, hbC] with ω hk hcω
intro t
exact (abs_mul _ _).trans_le (mul_le_mul (hk t) (hcω t) (abs_nonneg _) hc)
obtain ⟨L, hLp, hL, hcL, hL2, _, _, _, _, hLlim⟩ :=
exists_continuous_stopped_uniform_integral hB hAd hind hcB hKC hcKC
(mul_nonneg hc ha) hbKC T
have hLterminal : Tendsto (fun n => ∫ ω,
(uniformSum B (fun t ω => K t ω * C t ω) T n ω - L T ω) ^ 2 ∂P)
atTop (𝓝 0) := by
simpa only [stoppedUniformSum_terminal] using hLlim T
have houter := BrownianContinuousAssociativity.uniformSum_associativity
hB hAd hind hC hK hcK ha hc hbC hbK hI hI2 T hIlim (hL2 T) hLterminal
have hIright := martingale_rightCont_of_ae_continuous_memLp_two hI hI2 hcI
have hJright := martingale_rightCont_of_ae_continuous_memLp_two hJ hJ2 hcJ
have hLright := martingale_rightCont_of_ae_continuous_memLp_two hL hL2 hcL
have hKright : StronglyAdapted F.rightCont K :=
fun t => (hK t).mono (F.le_rightCont t)
have hstop := StoppedIntegralAssociativity.tendsto_stopped_outer_uniformSum
hIright hI2 hcI hKright hbK hLright hL2 hcL T houter hτ
have hBM := BrownianFiltration.martingale_of_independent_past hB hAd hind
have hB2 (t : ℝ≥0) := (hB.isGaussianProcess.hasGaussianLaw_eval t).memLp_two
have hBright := martingale_rightCont_of_ae_continuous_memLp_two hBM hB2 hcB
have heq := StoppedIntegralLocality.stopped_integral_eq_of_local_eq
hBright hB2 hcB (fun t => (hKC t).mono (F.le_rightCont t))
(fun t => (hR t).mono (F.le_rightCont t)) hbKC hbR
hLright hJright hL2 hJ2 hcL hcJ T hLterminal hJlim hτ hlocal
apply hstop.congr
intro n
apply integral_congr_ae
filter_upwards [heq] with ω hω
rw [hω]
theorem stopped_row_normalization {ι : Type*} [IsProbabilityMeasure P]
{F : Filtration ℝ≥0 mΩ} {B C R I J : ι → ℝ≥0 → Ω → ℝ}
{M Z K : ℝ≥0 → Ω → ℝ}
(hB : ∀ a, IsPreBrownianReal (B a) P)
(hAd : ∀ a, StronglyAdapted F (B a))
(hind : ∀ a s t, s ≤ t → Indep
(MeasurableSpace.comap (fun ω => B a t ω - B a s ω) inferInstance) (F s) P)
(hcB : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => B a t ω))
(hC : ∀ a, StronglyAdapted F (C a)) (hR : ∀ a, StronglyAdapted F (R a))
(hK : StronglyAdapted F K)
(hcC : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => C a t ω))
(hcK : ∀ᵐ ω ∂P, Continuous (fun t => K t ω))
{c r : ι → ℝ} {k : ℝ} (hc : ∀ a, 0 ≤ c a) (hk : 0 ≤ k)
(hbC : ∀ a, ∀ᵐ ω ∂P, ∀ t, |C a t ω| ≤ c a)
(hbR : ∀ a, ∀ᵐ ω ∂P, ∀ t, |R a t ω| ≤ r a)
(hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ k)
(hI : ∀ a, Martingale (I a) F P) (hJ : ∀ a, Martingale (J a) F P)
(hM : Martingale M F P) (hZ : Martingale Z F P)
(hI2 : ∀ a t, MemLp (I a t) 2 P) (hJ2 : ∀ a t, MemLp (J a t) 2 P)
(hM2 : ∀ t, MemLp (M t) 2 P) (hZ2 : ∀ t, MemLp (Z t) 2 P)
(hcI : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => I a t ω))
(hcJ : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => J a t ω))
(hcM : ∀ᵐ ω ∂P, Continuous (fun t => M t ω))
(hcZ : ∀ᵐ ω ∂P, Continuous (fun t => Z t ω))
(hpI : ∀ a, IsStronglyProgressive F (I a)) (hpM : IsStronglyProgressive F M)
(T : ℝ≥0)
(hsumI : HasSum (fun a => (hI2 a T).toLp (I a T)) ((hM2 T).toLp (M T)))
(hsumJ : HasSum (fun a => (hJ2 a T).toLp (J a T)) ((hZ2 T).toLp (Z T)))
(hIlim : ∀ a, Tendsto (fun n => ∫ ω,
(uniformSum (B a) (C a) T n ω - I a T ω) ^ 2 ∂P) atTop (𝓝 0))
(hJlim : ∀ a, Tendsto (fun n => ∫ ω,
(uniformSum (B a) (R a) T n ω - J a T ω) ^ 2 ∂P) atTop (𝓝 0))
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F.rightCont τ)
(hlocal : ∀ a, ∀ᵐ ω ∂P, ∀ t : ℝ≥0, (t : WithTop ℝ≥0) ≤ τ ω →
K t ω * C a t ω = R a t ω) :
Tendsto (fun n => ∫ ω,
(uniformSum (stoppedProcess M τ) K T n ω - stoppedProcess Z τ T ω) ^ 2 ∂P)
atTop (𝓝 0) := by
have hIr (a : ι) := martingale_rightCont_of_ae_continuous_memLp_two (hI a) (hI2 a) (hcI a)
have hJr (a : ι) := martingale_rightCont_of_ae_continuous_memLp_two (hJ a) (hJ2 a) (hcJ a)
have hMr := martingale_rightCont_of_ae_continuous_memLp_two hM hM2 hcM
have hZr := martingale_rightCont_of_ae_continuous_memLp_two hZ hZ2 hcZ
have hIs (a : ι) := martingale_stoppedProcess_rightCont_of_memLp_two
(hI a) (hI2 a) (hpI a) (hcI a) hτ
have hMs := martingale_stoppedProcess_rightCont_of_memLp_two hM hM2 hpM hcM hτ
have hIs2 (a : ι) (t : ℝ≥0) :=
StoppedMartingaleL2.memLp_stoppedProcess_two (hIr a) (hI2 a) (hcI a) hτ t
have hMs2 (t : ℝ≥0) :=
StoppedMartingaleL2.memLp_stoppedProcess_two hMr hM2 hcM hτ t
obtain ⟨hIsT, hMsT, hsI⟩ :=
MartingaleSpatialRandomEvaluation.exists_hasSum_stoppedProcess hIr hMr hI2 hM2
hcI hcM T hsumI hτ (t := T) le_rfl
obtain ⟨hJsT, hZsT, hsJ⟩ :=
MartingaleSpatialRandomEvaluation.exists_hasSum_stoppedProcess hJr hZr hJ2 hZ2
hcJ hcZ T hsumJ hτ (t := T) le_rfl
apply BrownianSpatialAssociativity.uniformSum_spatial_limit_of_channel_limits
hIs hMs hIs2 hMs2 hJsT hZsT (fun t => (hK t).mono (F.le_rightCont t)) hk
(fun t => hbK.mono (fun ω hω => hω t)) T hsI hsJ
intro a
exact stopped_channel_normalization (hB a) (hAd a) (hind a) (hcB a)
(hC a) (hR a) hK (hcC a) hcK (hc a) hk (hbC a) (hbR a) hbK
(hI a) (hJ a) (hI2 a) (hJ2 a) (hcI a) (hcJ a) T (hIlim a) (hJlim a)
hτ (hlocal a)
end Problem367.StoppedRowNormalization
end
end OAI
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