| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367 |
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| variable {Omega : Type*} [MeasurableSpace Omega] |
| {B : ℝ≥0 → Omega → ℝ} {mu : Measure Omega} |
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| def brownianBridge (B : ℝ≥0 → Omega → ℝ) (T : ℝ≥0) |
| (s : ℝ≥0) (w : Omega) : ℝ := |
| B s w - ((s : ℝ) / (T : ℝ)) * B T w |
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| lemma preBrownian_bridge_indep_terminal (hB : IsPreBrownianReal B mu) |
| {T : ℝ≥0} (hT : T ≠ 0) : |
| IndepFun (fun w (s : Set.Iic T) => brownianBridge B T s w) (B T) mu := by |
| classical |
| have hind : IndepFun |
| (fun w (s : Set.Iic T) => brownianBridge B T s w) |
| (fun w (_ : Unit) => B T w) mu := by |
| apply IsGaussianProcess.indepFun_of_covariance_eq_zero |
| · apply hB.isGaussianProcess.of_isGaussianProcess |
| rintro (⟨s, hs⟩ | u) |
| · exact ⟨{s, T}, |
| { toFun x := x ⟨s, by simp⟩ - ((s : ℝ) / (T : ℝ)) * x ⟨T, by simp⟩ |
| map_add' x y := by simp; ring |
| map_smul' c x := by simp; ring }, by simp [brownianBridge]⟩ |
| · exact ⟨{T}, |
| { toFun x := x ⟨T, by simp⟩ |
| map_add' x y := by simp |
| map_smul' c x := by simp }, by simp⟩ |
| · intro s |
| exact (hB.aemeasurable s).sub ((hB.aemeasurable T).const_mul _) |
| · intro u |
| exact hB.aemeasurable T |
| · intro s u |
| have : IsProbabilityMeasure mu := hB.isGaussianProcess.isProbabilityMeasure |
| change cov[fun w => B s w - ((s : ℝ) / (T : ℝ)) * B T w, B T; mu] = 0 |
| rw [covariance_fun_sub_left, covariance_const_mul_left, |
| hB.covariance_eval, hB.covariance_eval, min_eq_left s.property, min_self] |
| · have hT' : (T : ℝ) ≠ 0 := by exact_mod_cast hT |
| simp [div_mul_cancel₀ _ hT'] |
| · exact (hB.isGaussianProcess.hasGaussianLaw_eval s).memLp_two |
| · exact (hB.isGaussianProcess.hasGaussianLaw_eval T).memLp_two.const_mul _ |
| · exact (hB.isGaussianProcess.hasGaussianLaw_eval T).memLp_two |
| simpa only [Function.comp_def, id_eq] using |
| hind.comp measurable_id (measurable_pi_apply ()) |
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| omit [MeasurableSpace Omega] |
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| lemma brownianBridge_add_drift (B : ℝ≥0 → Omega → ℝ) (sigma : Omega → ℝ) |
| {T : ℝ≥0} (hT : T ≠ 0) (s : ℝ≥0) (w : Omega) : |
| brownianBridge (fun t w => (t : ℝ) * sigma w + B t w) T s w = |
| brownianBridge B T s w := by |
| have hT' : (T : ℝ) ≠ 0 := by exact_mod_cast hT |
| simp only [brownianBridge] |
| field_simp |
| ring |
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| lemma comap_past_eq_endpoint_bridge (B : ℝ≥0 → Omega → ℝ) (T : ℝ≥0) : |
| MeasurableSpace.comap (fun w (s : Set.Iic T) => B s w) inferInstance = |
| MeasurableSpace.comap |
| (fun w => (B T w, fun s : Set.Iic T => brownianBridge B T s w)) inferInstance := by |
| apply le_antisymm |
| · let : MeasurableSpace Omega := MeasurableSpace.comap |
| (fun w => (B T w, fun s : Set.Iic T => brownianBridge B T s w)) inferInstance |
| apply Measurable.comap_le |
| have hm : Measurable |
| (fun w => (B T w, fun s : Set.Iic T => brownianBridge B T s w)) := |
| comap_measurable _ |
| apply Measurable.of_eval |
| intro s |
| convert ((measurable_pi_apply s).comp (measurable_snd.comp hm)).add |
| ((measurable_fst.comp hm).const_mul ((s : ℝ≥0) / (T : ℝ))) using 1 |
| funext w |
| simp [brownianBridge] |
| · let : MeasurableSpace Omega := |
| MeasurableSpace.comap (fun w (s : Set.Iic T) => B s w) inferInstance |
| apply Measurable.comap_le |
| have hm : Measurable (fun w (s : Set.Iic T) => B s w) := comap_measurable _ |
| have ht : Measurable (B T) := (measurable_pi_apply (⟨T, by simp⟩ : Set.Iic T)).comp hm |
| apply ht.prodMk |
| apply Measurable.of_eval |
| intro s |
| exact ((measurable_pi_apply s).comp hm).sub (ht.const_mul _) |
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| end Problem367 |
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| end |
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| end OAI |
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