| 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 | |