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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 | import OAI.Probability.FreeIsing.Brownian.IntegralProcess
import OAI.Probability.FreeIsing.Estimates.CompensatedSquareLimit
import OAI.Probability.FreeIsing.Stochastic.SelectedCompensator
import OAI.Probability.FreeIsing.Stochastic.StoppedBrownianSquare
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal ENNReal Topology
namespace Problem367.BrownianIntegralProcess
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
theorem stopped_uniform_limit_compensatedSquare
{B H M : 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)
(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) (hM : StronglyAdapted F M) (hM2 : ∀ t, MemLp (M t) 2 P)
(hlim : ∀ t, Tendsto (fun n => ∫ ω,
(stoppedUniformSum B H T n t ω - M t ω) ^ 2 ∂P) atTop (𝓝 0)) :
Martingale (fun t ω => (M t ω) ^ 2 - SelectedCompensator.selected H T t ω) F P := by
let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
have hH2 (t : ℝ≥0) : MemLp (H t) 2 P := by
apply MemLp.of_bound hH.stronglyMeasurable.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 hfinite (n : ℕ) : Martingale (fun t ω =>
(stoppedUniformSum B H T n t ω) ^ 2 - SelectedCompensator.squareSum H T n t ω) F P := by
have h := StoppedBrownianSquare.stopped_predictable_sum_square_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 => hb.mono (fun ω hω => by simpa only [Real.norm_eq_abs] using hω _)) n
simpa only [stoppedUniformSum, SelectedCompensator.squareSum, mul_comm] using h
apply CompensatedSquareLimit.martingale_compensatedSquare_of_limits (l := atTop)
hS2 (fun n t => (SelectedCompensator.squareSum_memLp hH hC hb T t n 1).integrable le_rfl)
hfinite hM hM2 (SelectedCompensator.selected_stronglyAdapted hH T)
(fun t => (SelectedCompensator.selected_memLp hH hc hC hb T t 1).integrable le_rfl)
· intro t
exact CompensatedSquareLimit.tendsto_eLpNorm_two_of_meanSquare
(fun n => hS2 n t) (hM2 t) (hlim t)
· intro t
exact SelectedCompensator.tendsto_squareSum_L1_selected
(fun s => hH.stronglyMeasurable.aestronglyMeasurable) hc hC hb T t
theorem exists_stopped_uniform_integral_compensatedSquare
{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)
(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) :
∃ M Q : ℝ≥0 → Ω → ℝ, Martingale M F P ∧
(∀ t, MemLp (M t) 2 P) ∧ StronglyAdapted F Q ∧
(∀ᵐ ω ∂P, Continuous (fun t => Q t ω)) ∧
(∀ᵐ ω ∂P, ∀ t, Q t ω =
∫ s in (0 : ℝ)..min (t : ℝ) T, (H s.toNNReal ω) ^ 2) ∧
Martingale (fun t ω => (M t ω) ^ 2 - Q t ω) F P ∧
(∀ t, Tendsto (fun n => ∫ ω,
(stoppedUniformSum B H T n t ω - M t ω) ^ 2 ∂P) atTop (𝓝 0)) := by
obtain ⟨M, hM, hM2, _, _, _, hlim⟩ :=
exists_stopped_uniform_integral_process hB hAd hind hH hc hC hb T
exact ⟨M, SelectedCompensator.selected H T, hM, hM2,
SelectedCompensator.selected_stronglyAdapted hH T,
SelectedCompensator.ae_continuous_selected hc T,
SelectedCompensator.ae_selected_eq_integral hc T,
stopped_uniform_limit_compensatedSquare hB hAd hind hH hc hC hb T hM.stronglyAdapted hM2 hlim,
hlim⟩
end Problem367.BrownianIntegralProcess
end
end OAI
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