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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 | import OAI.Probability.FreeIsing.Brownian.ContinuousAssociativity
import OAI.Probability.FreeIsing.Stochastic.RandomStoppedBrownianIntegral
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.StoppedIntegralAssociativity
open BrownianGrid BrownianIntegralProcess
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
theorem stoppedUniformSum_martingale [IsFiniteMeasure P]
{M K : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ}
(hM : Martingale M F P) (hK : StronglyAdapted F K)
{D : ℝ} (hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D) (T : ℝ≥0) (n : ℕ) :
Martingale (stoppedUniformSum M K T n) F P := by
apply StoppedSimpleIntegral.finite_intervalSum_martingale hM (Finset.range n)
(uniformGrid T n) (fun i => uniformGrid T n (i + 1))
(fun i => K (uniformGrid T n i))
(fun i _ => uniformGrid_mono T n (Nat.le_succ i))
(fun i _ => hK _) (fun _ => D)
intro i hi
filter_upwards [hbK] with ω hω
exact hω _
theorem stoppedUniformSum_memLp [IsFiniteMeasure P]
{M K : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ}
(hM2 : ∀ t, MemLp (M t) 2 P) (hK : StronglyAdapted F K)
{D : ℝ} (hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D)
(T : ℝ≥0) (n : ℕ) (t : ℝ≥0) :
MemLp (stoppedUniformSum M K T n t) 2 P := by
apply memLp_finsetSum
intro i hi
exact BrownianGrid.memLp_compose_of_bounded
((hM2 (min t (uniformGrid T n (i + 1)))).sub (hM2 (min t (uniformGrid T n i))))
((hK _).mono (F.le _)).aestronglyMeasurable (hbK.mono (fun ω hω => hω _))
theorem tendsto_stopped_outer_uniformSum [IsFiniteMeasure P]
{M K J : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ}
(hM : Martingale M F P) (hM2 : ∀ t, MemLp (M t) 2 P)
(hcM : ∀ᵐ ω ∂P, Continuous (fun t => M t ω))
(hK : StronglyAdapted F K) {D : ℝ}
(hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D)
(hJ : Martingale J F P) (hJ2 : ∀ t, MemLp (J t) 2 P)
(hcJ : ∀ᵐ ω ∂P, Continuous (fun t => J t ω)) (T : ℝ≥0)
(hlim : Tendsto (fun n => ∫ ω,
(uniformSum M K T n ω - J T ω) ^ 2 ∂P) atTop (𝓝 0))
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F τ) :
Tendsto (fun n => ∫ ω,
(uniformSum (stoppedProcess M τ) K T n ω - stoppedProcess J τ T ω) ^ 2 ∂P)
atTop (𝓝 0) := by
have hterminal : Tendsto (fun n => ∫ ω,
(stoppedUniformSum M K T n T ω - J T ω) ^ 2 ∂P) atTop (𝓝 0) := by
simpa only [stoppedUniformSum_terminal] using hlim
have h := StoppedMartingaleL2.tendsto_stoppedProcess_meanSquare
(N := fun n => stoppedUniformSum M K T n) (L := J) (t := T) (T := T)
(fun n => stoppedUniformSum_martingale hM hK hbK T n) hJ
(fun n t => stoppedUniformSum_memLp hM2 hK hbK T n t) hJ2
(ae_continuous_stoppedUniformSum hcM T) hcJ (fun _ => τ) (fun _ => hτ)
le_rfl hterminal
have heq (n : ℕ) : stoppedProcess (stoppedUniformSum M K T n) τ T =
uniformSum (stoppedProcess M τ) K T n := by
have hfinite := RandomStoppedSimpleIntegral.stopped_finite_intervalSum
(Finset.range n) M (fun i => K (uniformGrid T n i))
(uniformGrid T n) (fun i => uniformGrid T n (i + 1)) τ
have ht := congrArg (fun S => S T) hfinite
change stoppedProcess (stoppedUniformSum M K T n) τ T =
stoppedUniformSum (stoppedProcess M τ) K T n T at ht
simpa only [stoppedUniformSum_terminal] using ht
simpa only [heq] using h
theorem stopped_uniformSum_associativity
{B H K I J : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ}
(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)
(hH : StronglyAdapted F H) (hK : StronglyAdapted F K)
(hcK : ∀ᵐ ω ∂P, Continuous (fun t => K t ω))
{C D : ℝ} (hC : 0 ≤ C) (hD : 0 ≤ D)
(hbH : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C)
(hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D)
(hI : Martingale I F P) (hI2 : ∀ t, MemLp (I t) 2 P)
(hcI : ∀ᵐ ω ∂P, Continuous (fun t => I t ω))
(hJ : Martingale J F P) (hJ2 : ∀ t, MemLp (J t) 2 P)
(hcJ : ∀ᵐ ω ∂P, Continuous (fun t => J t ω)) (T : ℝ≥0)
(hIlim : Tendsto (fun n => ∫ ω,
(uniformSum B H T n ω - I T ω) ^ 2 ∂P) atTop (𝓝 0))
(hJlim : Tendsto (fun n => ∫ ω,
(uniformSum B (fun t ω => K t ω * H t ω) T n ω - J T ω) ^ 2 ∂P)
atTop (𝓝 0))
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F τ) :
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
exact tendsto_stopped_outer_uniformSum hI hI2 hcI hK hbK hJ hJ2 hcJ T
(BrownianContinuousAssociativity.uniformSum_associativity hB hAd hind hH hK hcK
hC hD hbH hbK hI hI2 T hIlim (hJ2 T) hJlim) hτ
end Problem367.StoppedIntegralAssociativity
end
end OAI
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