| import OAI.Probability.FreeIsing.Brownian.SpatialAssociativity |
| import OAI.Probability.FreeIsing.Stochastic.StoppedIntegralAssociativity |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
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| namespace Problem367.StoppedIntegralLocality |
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| variable {Ω : Type*} |
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| theorem uniformSum_stopped_congr (M H K : ℝ≥0 → Ω → ℝ) |
| (τ : Ω → WithTop ℝ≥0) (T : ℝ≥0) (n : ℕ) (ω : Ω) |
| (hlocal : ∀ t : ℝ≥0, (t : WithTop ℝ≥0) ≤ τ ω → H t ω = K t ω) : |
| BrownianGrid.uniformSum (stoppedProcess M τ) H T n ω = |
| BrownianGrid.uniformSum (stoppedProcess M τ) K T n ω := by |
| unfold BrownianGrid.uniformSum BrownianGrid.integralSum |
| apply Finset.sum_congr rfl |
| intro i hi |
| dsimp only |
| by_cases ht : (BrownianGrid.uniformGrid T n i : WithTop ℝ≥0) ≤ τ ω |
| · rw [hlocal _ ht] |
| · have ha : τ ω ≤ (BrownianGrid.uniformGrid T n i : WithTop ℝ≥0) := |
| le_of_not_ge ht |
| have hab : (BrownianGrid.uniformGrid T n i : WithTop ℝ≥0) ≤ |
| (BrownianGrid.uniformGrid T n (i + 1) : WithTop ℝ≥0) := by |
| exact_mod_cast BrownianGrid.uniformGrid_mono T n (Nat.le_succ i) |
| rw [stoppedProcess_eq_of_ge ha, stoppedProcess_eq_of_ge (ha.trans hab)] |
| simp only [sub_self, mul_zero] |
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| variable [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] |
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| theorem stopped_integral_eq_of_local_eq |
| {F : Filtration ℝ≥0 mΩ} {M H K I J : ℝ≥0 → Ω → ℝ} |
| (hM : Martingale M F P) (hM2 : ∀ t, MemLp (M t) 2 P) |
| (hcM : ∀ᵐ ω ∂P, Continuous (fun t => M t ω)) |
| (hH : StronglyAdapted F H) (hK : StronglyAdapted F K) |
| {C D : ℝ} (hbH : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C) |
| (hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D) |
| (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, |
| (BrownianGrid.uniformSum M H T n ω - I T ω) ^ 2 ∂P) atTop (𝓝 0)) |
| (hJlim : Tendsto (fun n => ∫ ω, |
| (BrownianGrid.uniformSum M K T n ω - J T ω) ^ 2 ∂P) atTop (𝓝 0)) |
| {τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F τ) |
| (hlocal : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, (t : WithTop ℝ≥0) ≤ τ ω → H t ω = K t ω) : |
| stoppedProcess I τ T =ᵐ[P] stoppedProcess J τ T := by |
| have hIs := StoppedMartingaleL2.memLp_stoppedProcess_two hI hI2 hcI hτ T |
| have hJs := StoppedMartingaleL2.memLp_stoppedProcess_two hJ hJ2 hcJ hτ T |
| have hMs (t : ℝ≥0) := |
| StoppedMartingaleL2.memLp_stoppedProcess_two hM hM2 hcM hτ t |
| have hX (n : ℕ) : MemLp (BrownianGrid.uniformSum (stoppedProcess M τ) H T n) 2 P := |
| BrownianSpatialAssociativity.uniformSum_memLp hMs |
| (fun t => ((hH t).mono (F.le t)).aestronglyMeasurable) |
| (fun t => hbH.mono (fun ω hω => hω t)) T n |
| have hi := StoppedIntegralAssociativity.tendsto_stopped_outer_uniformSum |
| hM hM2 hcM hH hbH hI hI2 hcI T hIlim hτ |
| have hj := StoppedIntegralAssociativity.tendsto_stopped_outer_uniformSum |
| hM hM2 hcM hK hbK hJ hJ2 hcJ T hJlim hτ |
| apply BrownianGrid.ae_eq_of_mean_square_limits hX hIs hJs hi |
| apply hj.congr |
| intro n |
| apply integral_congr_ae |
| filter_upwards [hlocal] with ω hω |
| rw [uniformSum_stopped_congr M H K τ T n ω hω] |
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| end Problem367.StoppedIntegralLocality |
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| end |
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| end OAI |
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