File size: 3,990 Bytes
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 | import OAI.Probability.FreeIsing.Estimates.IntegrableTimePrimitive
import OAI.Probability.FreeIsing.Martingales.StoppedTimePrimitive
import OAI.Probability.FreeIsing.Spectral.L2CompensatedSquareStopping
import OAI.Probability.FreeIsing.Trees.InnovationPathMap
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal Topology
namespace Problem367.L2CompensatedDensityStopping
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
{F : Filtration ℝ≥0 mΩ} {M q : ℝ≥0 → Ω → ℝ}
theorem integrable_density_on_horizon
(hq : ∀ N : ℕ, Integrable (fun p : ℝ × Ω => q p.1.toNNReal p.2)
((volume.restrict (Ioc (0 : ℝ) (N : ℝ))).prod μ))
(T : ℝ≥0) :
Integrable (fun p : ℝ × Ω => q p.1.toNNReal p.2)
((volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod μ) := by
obtain ⟨N, hN⟩ := exists_nat_ge (T : ℝ)
exact (hq N).mono_measure (Measure.prod_mono
(Measure.restrict_mono (Ioc_subset_Ioc le_rfl hN) le_rfl) le_rfl)
theorem integrable_stopped_density_on_horizon
(hprogq : IsStronglyProgressive F q)
(hq : ∀ N : ℕ, Integrable (fun p : ℝ × Ω => q p.1.toNNReal p.2)
((volume.restrict (Ioc (0 : ℝ) (N : ℝ))).prod μ))
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F.rightCont τ)
(T : ℝ≥0) :
Integrable (fun p : ℝ × Ω =>
StoppedTimePrimitive.stopDensity q τ p.1.toNNReal p.2)
((volume.restrict (Ioc (0 : ℝ) (T : ℝ))).prod μ) := by
have hp := StoppedTimePrimitive.isStronglyProgressive_stopDensity
(DominatedMartingaleRightContinuation.isStronglyProgressive_mono_filtration
F.le_rightCont hprogq) hτ
have hm : StronglyMeasurable (fun p : ℝ × Ω =>
StoppedTimePrimitive.stopDensity q τ p.1.toNNReal p.2) :=
(stronglyMeasurable_joint_of_isStronglyProgressive hp).comp_measurable
((measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd)
apply (integrable_density_on_horizon hq T).mono hm.aestronglyMeasurable
apply Eventually.of_forall
intro p
unfold StoppedTimePrimitive.stopDensity
split_ifs <;> simp only [le_refl, norm_zero, norm_nonneg]
theorem stopped_martingale_and_compensated_density
(hM : Martingale M F μ) (h2 : ∀ t, MemLp (M t) 2 μ)
(hcM : ∀ᵐ ω ∂μ, Continuous (fun t => M t ω))
(hprogM : IsStronglyProgressive F M)
(hprogq : IsStronglyProgressive F q)
(hq0 : ∀ᵐ ω ∂μ, ∀ t, 0 ≤ q t ω)
(hq : ∀ N : ℕ, Integrable (fun p : ℝ × Ω => q p.1.toNNReal p.2)
((volume.restrict (Ioc (0 : ℝ) (N : ℝ))).prod μ))
(hQ : Martingale (fun t ω => M t ω ^ 2 - nnrealTimePrimitive q t ω) F μ)
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F.rightCont τ) :
Martingale (stoppedProcess M τ) F.rightCont μ ∧
Martingale (fun t ω => stoppedProcess M τ t ω ^ 2 -
nnrealTimePrimitive (StoppedTimePrimitive.stopDensity q τ) t ω)
F.rightCont μ := by
have hA0 : ∀ᵐ ω ∂μ, ∀ t, 0 ≤ nnrealTimePrimitive q t ω :=
hq0.mono (fun ω hω t => nnrealTimePrimitive_nonneg hω t)
have hAmono : ∀ᵐ ω ∂μ, Monotone (fun t => nnrealTimePrimitive q t ω) :=
ae_monotone_nnrealTimePrimitive_of_joint hq hq0
have hAint (t : ℝ≥0) : Integrable (nnrealTimePrimitive q t) μ :=
nnrealTimePrimitive_integrable (integrable_density_on_horizon hq t)
have hcA : ∀ᵐ ω ∂μ, Continuous (fun t => nnrealTimePrimitive q t ω) :=
ae_continuous_nnrealTimePrimitive_of_joint hq
refine ⟨L2CompensatedSquareStopping.martingale_stoppedProcess_rightCont
hM h2 hcM hprogM hτ, ?_⟩
have hs := L2CompensatedSquareStopping.stopped_compensatedSquare_rightCont
hM h2 hcM hA0 hAmono hAint hcA hQ hprogM
(isStronglyProgressive_nnrealTimePrimitive hprogq) hτ
simpa only [StoppedTimePrimitive.stoppedProcess_primitive] using hs
end Problem367.L2CompensatedDensityStopping
end
end OAI
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