File size: 3,730 Bytes
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 | import OAI.Probability.FreeIsing.Brownian.ContinuousIntegral
import OAI.Probability.FreeIsing.Brownian.IntegralCompensator
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 exists_continuous_integral_with_compensator
{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) :
∃ M Q : ℝ≥0 → Ω → ℝ,
IsStronglyPredictable F M ∧ IsStronglyPredictable F Q ∧
Martingale M F P ∧ (∀ t, MemLp (M t) 2 P) ∧
(∀ᵐ ω ∂P, Continuous (fun t => M t ω)) ∧ M 0 =ᵐ[P] 0 ∧
(∀ᵐ ω ∂P, Continuous (fun t => Q t ω) ∧ Monotone (fun t => Q t ω)) ∧
Q 0 =ᵐ[P] 0 ∧ (∀ t, Integrable (Q t) P) ∧
(∀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 ∧
(∀ᵐ ω ∂P, ∀ s t, |Q t ω - Q s ω| ≤ C ^ 2 * |(t : ℝ) - s|) ∧
(∀ t, Tendsto (fun n => ∫ ω,
(stoppedUniformSum B H T n t ω - M t ω) ^ 2 ∂P) atTop (𝓝 0)) := by
let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
obtain ⟨M, hp, hM, hMc, hM2, hMzero, _, _, _, hlim⟩ :=
exists_continuous_stopped_uniform_integral hB hAd hind hBc hH hc hC hb T
let Q := PredictableMesh.repair (SelectedCompensator.selected H T)
have hQp : IsStronglyPredictable F Q :=
PredictableMesh.repair_isStronglyPredictable
(SelectedCompensator.selected_stronglyAdapted hH T)
have hQeq : ∀ᵐ ω ∂P, ∀ t, Q t ω = SelectedCompensator.selected H T t ω := by
filter_upwards [SelectedCompensator.ae_continuous_selected hc T] with ω hω
intro t
exact PredictableMesh.repair_eq_of_continuousOn (T := t) hω.continuousOn le_rfl
have hcomp := stopped_uniform_limit_compensatedSquare
hB hAd hind hH hc hC hb T hM.stronglyAdapted hM2 hlim
refine ⟨M, Q, hp, hQp, hM, hM2, hMc, hMzero, ?_, ?_, ?_, ?_, ?_, ?_, hlim⟩
· filter_upwards [hQeq, SelectedCompensator.ae_continuous_selected hc T, hc]
with ω heq hcont hcω
constructor
· exact hcont.congr (fun t => (heq t).symm)
· intro s t hst
change Q s ω ≤ Q t ω
rw [heq s, heq t]
exact SelectedCompensator.selected_monotone_of_continuous hcω T hst
· filter_upwards [hQeq] with ω hω
simp only [hω 0, SelectedCompensator.selected_zero, Pi.zero_apply]
· intro t
apply ((SelectedCompensator.selected_memLp hH hc hC hb T t 1).integrable le_rfl).congr
filter_upwards [hQeq] with ω hω
exact (hω t).symm
· filter_upwards [hQeq, SelectedCompensator.ae_selected_eq_integral hc T] with ω heq hω
intro t
exact (heq t).trans (hω t)
· apply hcomp.congr (fun t => ((hM.stronglyAdapted t).pow 2).sub (hQp.stronglyAdapted t))
intro t
filter_upwards [hQeq] with ω hω
change M t ω ^ 2 - SelectedCompensator.selected H T t ω = M t ω ^ 2 - Q t ω
rw [hω t]
· filter_upwards [hQeq, hc, hb] with ω heq hcω hbω
intro s t
rw [heq s, heq t]
exact SelectedCompensator.abs_selected_sub_le hcω hC hbω T s t
end Problem367.BrownianIntegralProcess
end
end OAI
|