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f61e547 | 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 | import OAI.Probability.FreeIsing.Brownian.IntegralContinuitySupport
import OAI.Probability.FreeIsing.Martingales.L2Closure
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal Topology
namespace Problem367.BrownianIntegralProcess
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
omit mΩ in
theorem stoppedUniformSum_terminal (B H : ℝ≥0 → Ω → ℝ) (T : ℝ≥0) (n : ℕ) :
stoppedUniformSum B H T n T = BrownianGrid.uniformSum B H T n := by
funext ω
unfold stoppedUniformSum BrownianGrid.uniformSum BrownianGrid.integralSum
apply Finset.sum_congr rfl
intro i hi
have hin : i < n := Finset.mem_range.mp hi
have hn : 0 < n := lt_of_le_of_lt (Nat.zero_le i) hin
have ha := BrownianGrid.uniformGrid_le_horizon T hn hin.le
have hb := BrownianGrid.uniformGrid_le_horizon T hn hin
simp only [StoppedSimpleIntegral.intervalSum, min_eq_right ha, min_eq_right hb]
theorem integral_sq_sub_tendsto_of_L2_tendsto
{X : ℕ → Ω → ℝ} {L : Ω → ℝ}
(hX : ∀ n, MemLp (X n) 2 P) (hL : MemLp L 2 P)
(hlim : Tendsto (fun n => (hX n).toLp (X n)) atTop (𝓝 (hL.toLp L))) :
Tendsto (fun n => ∫ ω, (X n ω - L ω) ^ 2 ∂P) atTop (𝓝 0) := by
have hdist := (tendsto_iff_dist_tendsto_zero.mp hlim).pow 2
simpa only [MartingaleSquareMaximal.dist_toLp_sq, zero_pow (by decide : 2 ≠ 0)] using hdist
theorem exists_continuous_stopped_uniform_integral
{B H : 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)
(hBc : ∀ᵐ ω ∂P, Continuous (fun t => B t ω))
(hH : StronglyAdapted F H)
(hc : ∀ᵐ ω ∂P, Continuous (fun t => H t ω))
{C : ℝ} (hC : 0 ≤ C) (hb : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C) (T : ℝ≥0) :
∃ I : ℝ≥0 → Ω → ℝ, IsStronglyPredictable F I ∧ Martingale I F P ∧
(∀ᵐ ω ∂P, Continuous (fun t => I t ω)) ∧
(∀ t, MemLp (I t) 2 P) ∧ I 0 =ᵐ[P] 0 ∧
(∀ t, (∫ ω, I t ω ∂P) = 0) ∧
(∀ᵐ ω ∂P, ∀ t, T ≤ t → I t ω = I T ω) ∧
(∫ ω, (I T ω) ^ 2 ∂P) =
(∫ t in (0 : ℝ)..(T : ℝ), ∫ ω, (H t.toNNReal ω) ^ 2 ∂P) ∧
(∀ t, Tendsto (fun n => ∫ ω,
(stoppedUniformSum B H T n t ω - I t ω) ^ 2 ∂P) atTop (𝓝 0)) := by
let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
have hH2 (t : ℝ≥0) : MemLp (H t) 2 P := by
apply MemLp.of_bound ((hH t).mono (F.le t)).aestronglyMeasurable C
filter_upwards [hb] with ω hω
simpa only [Real.norm_eq_abs] using hω t
have hS2 (n : ℕ) (t : ℝ≥0) : MemLp (stoppedUniformSum B H T n t) 2 P :=
BrownianFiltration.stopped_predictable_sum_memLp_two hB hind
(BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n)
(fun i => H (BrownianGrid.uniformGrid T n i)) (fun i => hH _) (fun i => hH2 _)
(Finset.range n) t
have hSM (n : ℕ) : Martingale (stoppedUniformSum B H T n) F P := by
apply BrownianFiltration.stopped_predictable_sum_martingale hB hAd hind
(BrownianGrid.uniformGrid T n) (BrownianGrid.uniformGrid_mono T n)
(fun i => H (BrownianGrid.uniformGrid T n i)) (fun i => hH _) (fun _ => C)
(fun i => ?_) (Finset.range n)
filter_upwards [hb] with ω hω
simpa only [Real.norm_eq_abs] using hω (BrownianGrid.uniformGrid T n i)
have hSC : CauchySeq (fun n => (hS2 n T).toLp (stoppedUniformSum B H T n T)) := by
have hclass : (fun n => (hS2 n T).toLp (stoppedUniformSum B H T n T)) =
BrownianFiltrationGrid.uniformSumLp hB hAd hind H hH hH2 T := by
funext n
apply Lp.ext
exact (hS2 n T).coeFn_toLp.trans
((stoppedUniformSum_terminal B H T n).eventuallyEq.trans
(BrownianFiltrationGrid.coe_uniformSumLp hB hAd hind H hH hH2 T n).symm)
rw [hclass]
exact BrownianFiltrationGrid.cauchySeq_uniformSumLp hB hAd hind H hH hH2 T
(fun ε hε => MeanSquareContinuity.uniform_integral_sq_sub_le
(fun t => (hH2 t).aestronglyMeasurable) hc hC hb T hε)
obtain ⟨L, hLp, hLc, hLlim⟩ :=
MartingaleSquareMaximal.exists_predictable_continuous_L2_limit hSM hS2
(ae_continuous_stoppedUniformSum hBc T) T hSC
obtain ⟨M, hM, hM2, hMzero, hMstop, hMiso, hMlim⟩ :=
exists_stopped_uniform_integral_process hB hAd hind hH hc hC hb T
have hLM (t : ℝ≥0) (ht : t ≤ T) : L t =ᵐ[P] M t := by
obtain ⟨hL2, hlim⟩ := hLlim t ht
exact BrownianGrid.ae_eq_of_mean_square_limits (fun n => hS2 n t) hL2 (hM2 t)
(integral_sq_sub_tendsto_of_L2_tendsto (fun n => hS2 n t) hL2 hlim) (hMlim t)
let R : ℝ≥0 → Ω → ℝ := fun t => L (min t T)
have hRad : StronglyAdapted F R := fun t =>
(hLp.stronglyAdapted (min t T)).mono (F.mono (min_le_left _ _))
have hRc : ∀ᵐ ω ∂P, Continuous (fun t => R t ω) := by
filter_upwards [hLc] with ω hω
exact hω.comp_continuous (continuous_id.min continuous_const) (fun t => min_le_right t T)
let I := PredictableMesh.repair R
have hIp : IsStronglyPredictable F I := PredictableMesh.repair_isStronglyPredictable hRad
have hIR : ∀ᵐ ω ∂P, ∀ t, I t ω = R t ω := by
filter_upwards [hRc] with ω hω
intro t
exact PredictableMesh.repair_eq_of_continuousOn (T := t) hω.continuousOn le_rfl
have hIM (t : ℝ≥0) : I t =ᵐ[P] M t := by
filter_upwards [hIR, hLM (min t T) (min_le_right _ _)] with ω hIω hω
rw [hIω t]
change L (min t T) ω = M t ω
rw [hω]
by_cases ht : t ≤ T
· rw [min_eq_left ht]
· rw [min_eq_right (le_of_not_ge ht), hMstop t (le_of_not_ge ht)]
refine ⟨I, hIp, hM.congr hIp.stronglyAdapted (fun t => (hIM t).symm), ?_,
(fun t => (hM2 t).ae_eq (hIM t).symm), ?_, ?_, ?_, ?_, ?_⟩
· filter_upwards [hIR, hRc] with ω hIω hω
exact hω.congr (fun t => (hIω t).symm)
· exact (hIM 0).trans (initial_zero_of_stopped_uniform_limit T (hM2 0) (hMlim 0))
· intro t
exact (integral_congr_ae (hIM t)).trans (hMzero t)
· filter_upwards [hIR] with ω hω
intro t ht
rw [hω t, hω T]
dsimp [R]
rw [min_eq_right ht, min_self]
· rw [← hMiso]
apply integral_congr_ae
filter_upwards [hIM T] with ω hω
rw [hω]
· intro t
apply (hMlim t).congr'
apply Filter.Eventually.of_forall
intro n
apply integral_congr_ae
filter_upwards [hIM t] with ω hω
rw [hω]
end Problem367.BrownianIntegralProcess
end
end OAI
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