openAI-math-dataset / lean /OAI /Probability /FreeIsing /Martingales /StoppedSpatialAssociativity.lean
| import OAI.Probability.FreeIsing.Brownian.SpatialAssociativity | |
| import OAI.Probability.FreeIsing.Stochastic.StoppedIntegralAssociativity | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped NNReal Topology | |
| namespace Problem367.StoppedSpatialAssociativity | |
| variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} | |
| theorem stopped_uniformSum_spatial_associativity [IsFiniteMeasure P] | |
| {F : Filtration ℝ≥0 mΩ} {B H I J : ι → ℝ≥0 → Ω → ℝ} | |
| {Y 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) | |
| (hH : ∀ a, StronglyAdapted F (H a)) (hK : StronglyAdapted F K) | |
| (hcK : ∀ᵐ ω ∂P, Continuous (fun t => K t ω)) | |
| {C : ι → ℝ} {D : ℝ} (hC : ∀ a, 0 ≤ C a) (hD : 0 ≤ D) | |
| (hbH : ∀ a, ∀ᵐ ω ∂P, ∀ t, |H a t ω| ≤ C a) | |
| (hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D) | |
| (hI : ∀ a, Martingale (I a) F P) (hY : Martingale Y F P) | |
| (hI2 : ∀ a t, MemLp (I a t) 2 P) (hY2 : ∀ t, MemLp (Y t) 2 P) | |
| (hJ2 : ∀ a t, MemLp (J a t) 2 P) | |
| (hZ : Martingale Z F P) (hZ2 : ∀ t, MemLp (Z t) 2 P) | |
| (hcY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω)) | |
| (hcZ : ∀ᵐ ω ∂P, Continuous (fun t => Z t ω)) | |
| (T : ℝ≥0) | |
| (hsumI : HasSum (fun a => (hI2 a T).toLp (I a T)) ((hY2 T).toLp (Y T))) | |
| (hsumJ : HasSum (fun a => (hJ2 a T).toLp (J a T)) ((hZ2 T).toLp (Z T))) | |
| (hIlim : ∀ a, Tendsto (fun n => ∫ ω, | |
| (BrownianGrid.uniformSum (B a) (H a) T n ω - I a T ω) ^ 2 ∂P) | |
| atTop (𝓝 0)) | |
| (hJlim : ∀ a, Tendsto (fun n => ∫ ω, | |
| (BrownianGrid.uniformSum (B a) (fun t ω => K t ω * H a t ω) T n ω - | |
| J a T ω) ^ 2 ∂P) atTop (𝓝 0)) | |
| {τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F τ) : | |
| Tendsto (fun n => ∫ ω, | |
| (BrownianGrid.uniformSum (stoppedProcess Y τ) K T n ω - | |
| stoppedProcess Z τ T ω) ^ 2 ∂P) atTop (𝓝 0) := by | |
| have hglobal : Tendsto (fun n => ∫ ω, | |
| (BrownianGrid.uniformSum Y K T n ω - Z T ω) ^ 2 ∂P) atTop (𝓝 0) := by | |
| apply BrownianSpatialAssociativity.uniformSum_spatial_limit_of_channel_limits | |
| hI hY hI2 hY2 (fun a => hJ2 a T) (hZ2 T) hK hD | |
| (fun t => hbK.mono (fun ω hω => hω t)) T hsumI hsumJ | |
| intro a | |
| exact BrownianContinuousAssociativity.uniformSum_associativity | |
| (hB a) (hAd a) (hind a) (hH a) hK hcK (hC a) hD (hbH a) hbK | |
| (hI a) (hI2 a) T (hIlim a) (hJ2 a T) (hJlim a) | |
| exact StoppedIntegralAssociativity.tendsto_stopped_outer_uniformSum | |
| hY hY2 hcY hK hbK hZ hZ2 hcZ T hglobal hτ | |
| end Problem367.StoppedSpatialAssociativity | |
| end | |
| end OAI | |