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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 | import OAI.Probability.FreeIsing.Brownian.RandomEvaluation
import OAI.Probability.FreeIsing.Martingales.StoppedMartingaleL2
import OAI.Probability.FreeIsing.Stochastic.RandomStoppedSimpleIntegral
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.BrownianIntegralProcess
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
theorem tendsto_random_stopped_uniformSum
{B H I : 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) {C : ℝ}
(hb : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C)
(hI : Martingale I F P) (hI2 : ∀ t, MemLp (I t) 2 P)
(hIc : ∀ᵐ ω ∂P, Continuous (fun t => I t ω))
{T : ℝ≥0}
(hlim : Tendsto (fun n => ∫ ω,
(stoppedUniformSum B H T n T ω - I T ω) ^ 2 ∂P) atTop (𝓝 0))
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F τ)
{t : ℝ≥0} (ht : t ≤ T) :
Tendsto (fun n => ∫ ω,
(stoppedUniformSum (stoppedProcess B τ) (stoppedProcess H τ) T n t ω -
stoppedProcess I τ t ω) ^ 2 ∂P) atTop (𝓝 0) := by
have h := stopped_uniform_limit_random_evaluation hB hAd hind hBc hH hb
hI hI2 hIc hlim (StoppedMartingaleL2.measurable_cappedTime hτ t)
(fun _ => (WithTop.untopA_le (min_le_left _ _)).trans ht)
change Tendsto (fun n => ∫ ω,
(stoppedProcess (stoppedUniformSum B H T n) τ t ω -
stoppedProcess I τ t ω) ^ 2 ∂P) atTop (𝓝 0) at h
simpa only [RandomStoppedSimpleIntegral.stopped_uniformSum] using h
end Problem367.BrownianIntegralProcess
end
end OAI
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