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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 | import OAI.Probability.FreeIsing.Estimates.ContinuousOptionalStopping
import OAI.Probability.FreeIsing.Martingales.StrongMaximal
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.ContinuousL2Stopping
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
[IsFiniteMeasure P] {F : Filtration ℝ≥0 mΩ}
theorem martingale_stoppedProcess
{M : ℝ≥0 → Ω → ℝ} (hM : Martingale M F P)
(h2 : ∀ t, MemLp (M t) 2 P) (hp : IsStronglyProgressive F M)
(hc : ∀ᵐ ω ∂P, Continuous (fun t => M t ω))
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F τ) :
Martingale (stoppedProcess M τ) F P := by
apply ContinuousOptionalStopping.martingale_stoppedProcess_of_continuous_dominated
hM hp hc _ hτ
intro T
refine ⟨fun ω => 1 + MartingaleStrongMaximal.maximalSquare M T ω,
(integrable_const 1).add
(MartingaleStrongMaximal.integrable_maximalSquare hM h2 hc T), ?_⟩
filter_upwards [MartingaleStrongMaximal.ae_all_sq_le_maximalSquare hM h2 hc T] with ω hω
intro t ht
have hs := hω t ht
nlinarith [sq_abs (M t ω), sq_nonneg (|M t ω| - 1 / 2)]
end Problem367.ContinuousL2Stopping
end
end OAI
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