| import OAI.Probability.FreeIsing.Martingales.RandomEvaluation |
| import OAI.Probability.FreeIsing.Martingales.SpatialLimit |
| import OAI.Probability.FreeIsing.Stochastic.IntegralAssociativityLimit |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter |
| open scoped NNReal Topology |
|
|
| namespace Problem367.MartingaleSpatialRandomEvaluation |
|
|
| variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] |
| {F : Filtration ℝ≥0 mΩ} {I : ι → ℝ≥0 → Ω → ℝ} {Y : ℝ≥0 → Ω → ℝ} |
|
|
| theorem tendsto_finiteSum_random_evaluation |
| (hI : ∀ a, Martingale (I a) F P) (hY : Martingale Y F P) |
| (hI2 : ∀ a t, MemLp (I a t) 2 P) (hY2 : ∀ t, MemLp (Y t) 2 P) |
| (hcI : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => I a t ω)) |
| (hcY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω)) |
| (T : ℝ≥0) |
| (hsum : HasSum (fun a => (hI2 a T).toLp (I a T)) ((hY2 T).toLp (Y T))) |
| (r : Finset ι → Ω → ℝ≥0) (hr : ∀ q, Measurable (r q)) |
| (hb : ∀ q ω, r q ω ≤ T) : |
| Tendsto (fun q : Finset ι => ∫ ω, |
| ((∑ a ∈ q, I a (r q ω) ω) - Y (r q ω) ω) ^ 2 ∂P) atTop (𝓝 0) := by |
| classical |
| have ht := PredictableL2.tendsto_integral_sq_error_of_hasSum |
| (fun a => I a T) (fun a => hI2 a T) (hY2 T) hsum |
| apply squeeze_zero (fun q => integral_nonneg (fun ω => sq_nonneg _)) |
| (fun q => ?_) (by simpa only [mul_zero] using ht.const_mul 4) |
| have hc : ∀ᵐ ω ∂P, Continuous |
| (fun t => (MartingaleSpatialLimit.finiteSum I q - Y) t ω) := by |
| filter_upwards [MartingaleSpatialLimit.finiteSum_ae_continuous hcI q, hcY] |
| with ω hq hy |
| exact hq.sub hy |
| simpa only [MartingaleSpatialLimit.finiteSum, Pi.sub_apply] using |
| MartingaleRandomEvaluation.integral_sq_random_evaluation_le |
| ((MartingaleSpatialLimit.finiteSum_martingale hI q).sub hY) |
| (fun t => (MartingaleSpatialLimit.finiteSum_memLp hI2 q t).sub (hY2 t)) |
| hc (hr q) (hb q) |
|
|
| theorem exists_hasSum_random_evaluation |
| (hI : ∀ a, Martingale (I a) F P) (hY : Martingale Y F P) |
| (hI2 : ∀ a t, MemLp (I a t) 2 P) (hY2 : ∀ t, MemLp (Y t) 2 P) |
| (hcI : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => I a t ω)) |
| (hcY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω)) |
| (T : ℝ≥0) |
| (hsum : HasSum (fun a => (hI2 a T).toLp (I a T)) ((hY2 T).toLp (Y T))) |
| (r : Ω → ℝ≥0) (hr : Measurable r) (hb : ∀ ω, r ω ≤ T) : |
| ∃ (hIr : ∀ a, MemLp (fun ω => I a (r ω) ω) 2 P) |
| (hYr : MemLp (fun ω => Y (r ω) ω) 2 P), |
| HasSum (fun a => (hIr a).toLp (fun ω => I a (r ω) ω)) |
| (hYr.toLp (fun ω => Y (r ω) ω)) := by |
| classical |
| have hIr (a : ι) : MemLp (fun ω => I a (r ω) ω) 2 P := |
| MartingaleRandomEvaluation.memLp_random_evaluation (hI a) (hI2 a) (hcI a) hr hb |
| have hYr : MemLp (fun ω => Y (r ω) ω) 2 P := |
| MartingaleRandomEvaluation.memLp_random_evaluation hY hY2 hcY hr hb |
| refine ⟨hIr, hYr, ?_⟩ |
| have hl := (SimpleIntegral.tendsto_toLp_iff_meanSquare |
| (fun q : Finset ι => memLp_finsetSum q (fun a _ => hIr a)) hYr).mpr |
| (tendsto_finiteSum_random_evaluation hI hY hI2 hY2 hcI hcY T hsum |
| (fun _ => r) (fun _ => hr) (fun _ => hb)) |
| change Tendsto (fun q : Finset ι => ∑ a ∈ q, (hIr a).toLp (fun ω => I a (r ω) ω)) |
| atTop (𝓝 (hYr.toLp (fun ω => Y (r ω) ω))) |
| exact hl.congr (fun q => PredictableL2.toLp_finsetSum |
| (fun a ω => I a (r ω) ω) hIr q) |
|
|
| theorem exists_hasSum_stoppedProcess |
| (hI : ∀ a, Martingale (I a) F P) (hY : Martingale Y F P) |
| (hI2 : ∀ a t, MemLp (I a t) 2 P) (hY2 : ∀ t, MemLp (Y t) 2 P) |
| (hcI : ∀ a, ∀ᵐ ω ∂P, Continuous (fun t => I a t ω)) |
| (hcY : ∀ᵐ ω ∂P, Continuous (fun t => Y t ω)) |
| (T : ℝ≥0) |
| (hsum : HasSum (fun a => (hI2 a T).toLp (I a T)) ((hY2 T).toLp (Y T))) |
| {τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F τ) |
| {t : ℝ≥0} (ht : t ≤ T) : |
| ∃ (hIs : ∀ a, MemLp (stoppedProcess (I a) τ t) 2 P) |
| (hYs : MemLp (stoppedProcess Y τ t) 2 P), |
| HasSum (fun a => (hIs a).toLp (stoppedProcess (I a) τ t)) |
| (hYs.toLp (stoppedProcess Y τ t)) := by |
| exact exists_hasSum_random_evaluation hI hY hI2 hY2 hcI hcY T hsum |
| (fun ω => (min (t : WithTop ℝ≥0) (τ ω)).untopA) |
| (WithTop.measurable_untopA.comp |
| (measurable_const.min (hτ.measurable.mono hτ.measurableSpace_le le_rfl))) |
| (fun ω => (WithTop.untopA_le (min_le_left _ _)).trans ht) |
|
|
| end Problem367.MartingaleSpatialRandomEvaluation |
|
|
| end |
|
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| end OAI |
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|
