| import OAI.Probability.FreeIsing.Estimates.ProgressivePrimitive |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped NNReal Topology |
| namespace Problem367.StoppedTimePrimitive |
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| def stopDensity {Ω : Type*} (a : ℝ≥0 → Ω → ℝ) (τ : Ω → WithTop ℝ≥0) |
| (t : ℝ≥0) (ω : Ω) : ℝ := |
| if (t : WithTop ℝ≥0) ≤ τ ω then a t ω else 0 |
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| theorem integral_indicator_min (a : ℝ → ℝ) (t u : ℝ≥0) : |
| (∫ r in (0 : ℝ)..(t : ℝ), if r ≤ u then a r else 0) = |
| ∫ r in (0 : ℝ)..(min t u : ℝ≥0), a r := by |
| rcases le_total t u with h | h |
| · rw [min_eq_left h] |
| apply intervalIntegral.integral_congr |
| intro r hr |
| rw [Set.uIcc_of_le t.coe_nonneg] at hr |
| exact ite_eq_left (hr.2.trans h) |
| · rw [min_eq_right h] |
| exact intervalIntegral.integral_indicator ⟨u.coe_nonneg, h⟩ |
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|
| theorem stoppedProcess_primitive {Ω : Type*} (a : ℝ≥0 → Ω → ℝ) |
| (τ : Ω → WithTop ℝ≥0) (t : ℝ≥0) (ω : Ω) : |
| stoppedProcess (nnrealTimePrimitive a) τ t ω = |
| nnrealTimePrimitive (stopDensity a τ) t ω := by |
| cases hτ : τ ω with |
| | top => simp only [stoppedProcess, nnrealTimePrimitive, stopDensity, hτ, min_eq_left le_top, WithTop.untopA, WithTop.untopD_coe, le_top, ite_true] |
| | coe u => |
| simp only [stoppedProcess, hτ, ← WithTop.coe_min, WithTop.untopA, |
| WithTop.untopD_coe, nnrealTimePrimitive, stopDensity, WithTop.coe_le_coe, |
| Real.toNNReal_le_iff_le_coe] |
| exact (integral_indicator_min (fun r => a r.toNNReal ω) t u).symm |
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| theorem continuous_stoppedProcess {Ω E : Type*} [TopologicalSpace E] |
| {X : ℝ≥0 → Ω → E} {ω : Ω} (hX : Continuous (fun t => X t ω)) |
| (τ : Ω → WithTop ℝ≥0) : Continuous (fun t => stoppedProcess X τ t ω) := by |
| cases hτ : τ ω with |
| | top => simpa only [stoppedProcess, hτ, min_eq_left le_top, WithTop.untopA, WithTop.untopD_coe] using hX |
| | coe u => |
| simpa only [stoppedProcess, hτ, ← WithTop.coe_min, WithTop.untopA, |
| WithTop.untopD_coe, Function.comp_def, id_eq] using hX.comp (continuous_id.min continuous_const) |
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| theorem monotone_stoppedProcess {Ω E : Type*} [Preorder E] |
| {X : ℝ≥0 → Ω → E} {ω : Ω} (hX : Monotone (fun t => X t ω)) |
| (τ : Ω → WithTop ℝ≥0) : Monotone (fun t => stoppedProcess X τ t ω) := by |
| cases hτ : τ ω with |
| | top => simpa only [stoppedProcess, hτ, min_eq_left le_top, WithTop.untopA, WithTop.untopD_coe] using hX |
| | coe u => |
| simpa only [stoppedProcess, hτ, ← WithTop.coe_min, WithTop.untopA, |
| WithTop.untopD_coe, Function.comp_def, id_eq] using hX.comp (monotone_id.min monotone_const) |
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| theorem stopTime_eq_iff_le {Ω : Type*} (τ : Ω → WithTop ℝ≥0) (t : ℝ≥0) (ω : Ω) : |
| (min (t : WithTop ℝ≥0) (τ ω)).untopA = t ↔ (t : WithTop ℝ≥0) ≤ τ ω := by |
| cases hτ : τ ω with |
| | top => simp only [min_eq_left le_top, WithTop.untopA, WithTop.untopD_coe, le_top] |
| | coe u => simp only [← WithTop.coe_min, WithTop.untopA, WithTop.untopD_coe, min_eq_left_iff, WithTop.coe_le_coe] |
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| theorem isStronglyProgressive_stopDensity {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| {F : Filtration ℝ≥0 mΩ} {a : ℝ≥0 → Ω → ℝ} {τ : Ω → WithTop ℝ≥0} |
| (ha : IsStronglyProgressive F a) (hτ : IsStoppingTime F τ) : |
| IsStronglyProgressive F (stopDensity a τ) := by |
| intro T |
| let : MeasurableSpace Ω := F T |
| have hs : MeasurableSet {p : Iic T × Ω | ((p.1 : ℝ≥0) : WithTop ℝ≥0) ≤ τ p.2} := by |
| simp only [← stopTime_eq_iff_le] |
| exact measurableSet_eq_fun (isStronglyProgressive_min_stopping_time hτ T).measurable |
| measurable_fst.subtype_val |
| exact (ha T).piecewise hs stronglyMeasurable_const |
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| theorem stoppedProcess_weighted_primitive {Ω E : Type*} |
| (X : ℝ≥0 → Ω → E) (f : E → ℝ) (a : ℝ≥0 → Ω → ℝ) |
| (τ : Ω → WithTop ℝ≥0) (t : ℝ≥0) (ω : Ω) : |
| stoppedProcess (nnrealTimePrimitive (fun s ω => f (X s ω) * a s ω)) τ t ω = |
| nnrealTimePrimitive (fun s ω => f (stoppedProcess X τ s ω) * stopDensity a τ s ω) t ω := by |
| rw [stoppedProcess_primitive] |
| congr 1 |
| funext s ω |
| by_cases h : (s : WithTop ℝ≥0) ≤ τ ω |
| · simp only [stopDensity, h, ite_true, stoppedProcess_eq_of_le h] |
| · simp only [stopDensity, h, ite_false, mul_zero] |
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| end Problem367.StoppedTimePrimitive |
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| end |
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| end OAI |
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