File size: 3,664 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 | import OAI.Probability.FreeIsing.Brownian.FiltrationSquare
import OAI.Probability.FreeIsing.Estimates.ContinuousOptionalStopping
import OAI.Probability.FreeIsing.Martingales.StrongMaximal
import OAI.Probability.FreeIsing.Stochastic.StoppedBrownianIntegral
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.ContinuousOptionalStopping
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
{F : Filtration ℝ≥0 mΩ} {M A : ℝ≥0 → Ω → ℝ} {τ : Ω → WithTop ℝ≥0}
theorem martingale_stopped_compensatedSquare_of_memLp_two
(hM : Martingale M F μ) (hM2 : ∀ t, MemLp (M t) 2 μ)
(hcomp : Martingale (fun t ω => M t ω ^ 2 - A t ω) F μ)
(hprogM : IsStronglyProgressive F M) (hprogA : IsStronglyProgressive F A)
(hcM : ∀ᵐ ω ∂μ, Continuous (fun t => M t ω))
(hcA : ∀ᵐ ω ∂μ, Continuous (fun t => A t ω))
(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)
(hτ : IsStoppingTime F τ) :
Martingale (fun t ω => stoppedProcess M τ t ω ^ 2 - stoppedProcess A τ t ω) F 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
apply martingale_stoppedProcess_of_continuous_dominated hcomp hprog hc _ hτ
intro T
refine ⟨fun ω => MartingaleStrongMaximal.maximalSquare M T ω + A T ω,
(MartingaleStrongMaximal.integrable_maximalSquare hM hM2 hcM T).add (hAint T), ?_⟩
filter_upwards [MartingaleStrongMaximal.ae_all_sq_le_maximalSquare hM hM2 hcM T,
hA0, hAmono] with ω hmax hpos hmono
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 (hpos t)]
_ ≤ _ := add_le_add (hmax t ht) (hmono ht)
end Problem367.ContinuousOptionalStopping
namespace Problem367.BrownianFiltration
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
{F : Filtration ℝ≥0 mΩ} {B : ℝ≥0 → Ω → ℝ} {τ : Ω → WithTop ℝ≥0}
theorem stopped_square_sub_clock_martingale
(hB : IsPreBrownianReal B P) (hprog : IsStronglyProgressive F B)
(hind : ∀ s t, s ≤ t → Indep
(MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P)
(hc : ∀ᵐ ω ∂P, Continuous (fun t => B t ω)) (hτ : IsStoppingTime F τ) :
Martingale (fun t ω => stoppedProcess B τ t ω ^ 2 -
((min (t : WithTop ℝ≥0) (τ ω)).untopA : ℝ)) F P := by
let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
have htime : IsStronglyProgressive F (fun t (_ : Ω) => (t : ℝ)) := by
intro t
let : MeasurableSpace Ω := F t
apply Measurable.stronglyMeasurable
fun_prop
exact ContinuousOptionalStopping.martingale_stopped_compensatedSquare_of_memLp_two
(martingale_of_independent_past hB hprog.stronglyAdapted hind)
(fun t => (hB.isGaussianProcess.hasGaussianLaw_eval t).memLp_two)
(square_sub_time_martingale hB hprog.stronglyAdapted hind) hprog htime hc
(Eventually.of_forall (fun ω => NNReal.continuous_coe))
(Eventually.of_forall (fun ω t => t.coe_nonneg))
(Eventually.of_forall (fun ω s t hst => by exact_mod_cast hst))
(fun t => integrable_const (t : ℝ)) hτ
end Problem367.BrownianFiltration
end
end OAI
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