| import OAI.Probability.FreeIsing.Martingales.L2MartingaleRightContinuation |
| import OAI.Probability.FreeIsing.Martingales.SpatialRandomEvaluation |
| import OAI.Probability.FreeIsing.Stochastic.StoppedIntegralLocality |
|
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| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
|
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| namespace Problem367.StoppedRowNormalization |
| open BrownianGrid BrownianIntegralProcess DominatedMartingaleRightContinuation |
|
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
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| 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] |
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| 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 |
|
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| end |
|
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| end OAI |
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