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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 | import OAI.Probability.FreeIsing.Martingales.DominatedMartingaleRightContinuation
import OAI.Probability.FreeIsing.Martingales.StrongMaximal
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.L2CompensatedSquareStopping
open MartingaleStrongMaximal DominatedMartingaleRightContinuation
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
{F : Filtration ℝ≥0 mΩ} {M A : ℝ≥0 → Ω → ℝ}
(hM : Martingale M F μ) (h2 : ∀ t, MemLp (M t) 2 μ)
(hcM : ∀ᵐ ω ∂μ, Continuous (fun t => M t ω))
include hM h2 hcM
theorem exists_integrable_horizon_envelope (T : ℝ≥0) :
∃ g : Ω → ℝ, Integrable g μ ∧
∀ᵐ ω ∂μ, ∀ t ≤ T, |M t ω| ≤ g ω := by
refine ⟨fun ω => maximalSquare M T ω + 1,
(integrable_maximalSquare hM h2 hcM T).add (integrable_const _), ?_⟩
filter_upwards [ae_all_sq_le_maximalSquare hM h2 hcM T] with ω hω
intro t ht
have habs : |M t ω| ≤ M t ω ^ 2 + 1 := by
nlinarith [sq_nonneg (|M t ω| - 1 / 2), sq_abs (M t ω)]
exact habs.trans (add_le_add (hω t ht) le_rfl)
theorem martingale_rightCont : Martingale M F.rightCont μ :=
martingale_rightCont_of_ae_continuous_dominated hM hcM
(exists_integrable_horizon_envelope hM h2 hcM)
theorem martingale_stoppedProcess_rightCont
(hprogM : IsStronglyProgressive F M)
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F.rightCont τ) :
Martingale (stoppedProcess M τ) F.rightCont μ :=
DominatedMartingaleRightContinuation.martingale_stoppedProcess_rightCont
hM hprogM hcM (exists_integrable_horizon_envelope hM h2 hcM) hτ
variable
(hA0 : ∀ᵐ ω ∂μ, ∀ t, 0 ≤ A t ω)
(hAmono : ∀ᵐ ω ∂μ, Monotone (fun t => A t ω))
(hAint : ∀ t, Integrable (A t) μ)
include hA0 hAmono hAint
theorem exists_integrable_compensatedSquare_envelope (T : ℝ≥0) :
∃ g : Ω → ℝ, Integrable g μ ∧
∀ᵐ ω ∂μ, ∀ t ≤ T, |M t ω ^ 2 - A t ω| ≤ g ω := by
refine ⟨fun ω => maximalSquare M T ω + A T ω,
(integrable_maximalSquare hM h2 hcM T).add (hAint T), ?_⟩
filter_upwards [ae_all_sq_le_maximalSquare hM h2 hcM T, hA0, hAmono]
with ω hω ha hm
intro t ht
calc
|M t ω ^ 2 - A t ω| ≤ |M t ω ^ 2| + |A t ω| := abs_sub _ _
_ = M t ω ^ 2 + A t ω := by
rw [abs_of_nonneg (sq_nonneg _), abs_of_nonneg (ha t)]
_ ≤ maximalSquare M T ω + A T ω := add_le_add (hω t ht) (hm ht)
variable (hcA : ∀ᵐ ω ∂μ, Continuous (fun t => A t ω))
(hQ : Martingale (fun t ω => M t ω ^ 2 - A t ω) F μ)
include hcA hQ
theorem compensatedSquare_martingale_rightCont :
Martingale (fun t ω => M t ω ^ 2 - A t ω) F.rightCont μ := by
apply martingale_rightCont_of_ae_continuous_dominated hQ
(show ∀ᵐ ω ∂μ, Continuous (fun t => M t ω ^ 2 - A t ω) from ?_)
(exists_integrable_compensatedSquare_envelope hM h2 hcM hA0 hAmono hAint)
filter_upwards [hcM, hcA] with ω hm ha
exact (hm.pow 2).sub ha
theorem stopped_compensatedSquare_rightCont
(hprogM : IsStronglyProgressive F M) (hprogA : IsStronglyProgressive F A)
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F.rightCont τ) :
Martingale (fun t ω => stoppedProcess M τ t ω ^ 2 - stoppedProcess A τ t ω)
F.rightCont μ := by
have hprog : IsStronglyProgressive F (fun t ω => M t ω ^ 2 - A t ω) := by
intro t
exact ((hprogM t).pow 2).sub (hprogA t)
have hc : ∀ᵐ ω ∂μ, Continuous (fun t => M t ω ^ 2 - A t ω) := by
filter_upwards [hcM, hcA] with ω hm ha
exact (hm.pow 2).sub ha
exact DominatedMartingaleRightContinuation.martingale_stoppedProcess_rightCont
hQ hprog hc
(exists_integrable_compensatedSquare_envelope hM h2 hcM hA0 hAmono hAint) hτ
end Problem367.L2CompensatedSquareStopping
end
end OAI
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