| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367.BrownianLinearCombination |
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| variable {ι Ω : Type*} [Fintype ι] [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {B : ι → ℝ≥0 → Ω → ℝ} |
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| omit [IsProbabilityMeasure μ] in |
|
|
| theorem isGaussianProcess_sum |
| (hB : ∀ i, IsGaussianProcess (B i) μ) |
| (hind : iIndepFun (fun i ω t => B i t ω) μ) (a : ι → ℝ) : |
| IsGaussianProcess (fun t ω => ∑ i, a i * B i t ω) μ := by |
| classical |
| constructor |
| intro I |
| let C : ι → Ω → (I → ℝ) := fun i ω t => a i * B i t ω |
| have hC (i : ι) : HasGaussianLaw (C i) μ := by |
| exact ((hB i).smul (fun _ => a i)).hasGaussianLaw I |
| have hCi : iIndepFun C μ := by |
| have h := hind.comp (fun i (x : ℝ≥0 → ℝ) (t : I) => a i * x t) |
| (fun i => by fun_prop) |
| exact h |
| have hsum := hCi.hasGaussianLaw_fun_sum hC |
| have heq : (fun ω => ∑ i, C i ω) = |
| (fun ω => I.restrict (fun t => ∑ i, a i * B i t ω)) := by |
| funext ω t |
| simp only [Finset.sum_apply, C, Finset.restrict_def] |
| rw [heq] at hsum |
| exact hsum |
|
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| omit [Fintype ι] [IsProbabilityMeasure μ] in |
|
|
| theorem covariance_channel_ne |
| (hB : ∀ i, IsPreBrownianReal (B i) μ) |
| (hind : iIndepFun (fun i ω t => B i t ω) μ) |
| {i j : ι} (hij : i ≠ j) (s t : ℝ≥0) : |
| cov[B i s, B j t; μ] = 0 := by |
| have h := (hind.indepFun hij).comp (measurable_pi_apply s) (measurable_pi_apply t) |
| exact h.covariance_eq_zero |
| ((hB i).isGaussianProcess.hasGaussianLaw_eval s).memLp_two |
| ((hB j).isGaussianProcess.hasGaussianLaw_eval t).memLp_two |
|
|
| theorem covariance_sum |
| (hB : ∀ i, IsPreBrownianReal (B i) μ) |
| (hind : iIndepFun (fun i ω t => B i t ω) μ) (a : ι → ℝ) |
| (s t : ℝ≥0) : |
| cov[fun ω => ∑ i, a i * B i s ω, fun ω => ∑ i, a i * B i t ω; μ] = |
| (∑ i, (a i) ^ 2) * (min s t : ℝ≥0) := by |
| classical |
| rw [covariance_fun_sum_fun_sum |
| (fun i => ((hB i).isGaussianProcess.hasGaussianLaw_eval s).memLp_two.const_mul (a i)) |
| (fun i => ((hB i).isGaussianProcess.hasGaussianLaw_eval t).memLp_two.const_mul (a i))] |
| have heq (i j : ι) : cov[fun ω => a i * B i s ω, fun ω => a j * B j t ω; μ] = |
| if i = j then (a i) ^ 2 * (min s t : ℝ≥0) else 0 := by |
| rw [covariance_const_mul_left, covariance_const_mul_right] |
| by_cases hij : i = j |
| · subst j |
| rw [ite_eq_left rfl, (hB i).covariance_eval] |
| ring |
| · rw [ite_eq_right hij, covariance_channel_ne hB hind hij, mul_zero, mul_zero] |
| simp_rw [heq] |
| simp [Finset.sum_mul] |
|
|
| theorem isPreBrownianReal_sum |
| (hB : ∀ i, IsPreBrownianReal (B i) μ) |
| (hind : iIndepFun (fun i ω t => B i t ω) μ) (a : ι → ℝ) |
| (ha : ∑ i, (a i) ^ 2 = 1) : |
| IsPreBrownianReal (fun t ω => ∑ i, a i * B i t ω) μ := by |
| apply (isGaussianProcess_sum (fun i => (hB i).isGaussianProcess) hind a).isPreBrownianReal_of_covariance |
| · intro t |
| rw [integral_finsetSum _ (fun i _ => ((hB i).integrable_eval t).const_mul (a i))] |
| simp_rw [integral_const_mul, IsPreBrownianReal.integral_eval (hB _), mul_zero] |
| simp |
| · intro s t hst |
| rw [covariance_sum hB hind a, ha, one_mul, min_eq_left hst] |
|
|
| theorem isBrownianReal_sum |
| (hB : ∀ i, IsBrownianReal (B i) μ) |
| (hind : iIndepFun (fun i ω t => B i t ω) μ) (a : ι → ℝ) |
| (ha : ∑ i, (a i) ^ 2 = 1) : |
| IsBrownianReal (fun t ω => ∑ i, a i * B i t ω) μ := by |
| refine ⟨isPreBrownianReal_sum (fun i => (hB i).toIsPreBrownianReal) hind a ha, ?_⟩ |
| filter_upwards [ae_all_iff.mpr (fun i => (hB i).cont)] with ω hω |
| exact continuous_finsetSum _ (fun i _ => continuous_const.mul (hω i)) |
|
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| theorem isPreBrownianReal_normalized_sum |
| (hB : ∀ i, IsPreBrownianReal (B i) μ) |
| (hind : iIndepFun (fun i ω t => B i t ω) μ) (a : ι → ℝ) |
| (ha : 0 < ∑ i, (a i) ^ 2) : |
| IsPreBrownianReal |
| (fun t ω => (Real.sqrt (∑ i, (a i) ^ 2))⁻¹ * ∑ i, a i * B i t ω) μ := by |
| have hnorm : ∑ i, (a i / Real.sqrt (∑ j, (a j) ^ 2)) ^ 2 = 1 := by |
| simp_rw [div_pow] |
| rw [← Finset.sum_div, Real.sq_sqrt ha.le, div_self ha.ne'] |
| have h := isPreBrownianReal_sum hB hind |
| (fun i => a i / Real.sqrt (∑ j, (a j) ^ 2)) hnorm |
| convert h using 1 |
| funext t ω |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| simp only [div_eq_mul_inv] |
| ring |
| |
| omit [IsProbabilityMeasure μ] in |
| |
| theorem ae_abs_sum_le {A : ι → Ω → ℝ} (a : ι → ℝ) {R : ℝ} |
| (hA : ∀ i, ∀ᵐ ω ∂μ, |A i ω| ≤ R) : |
| ∀ᵐ ω ∂μ, |∑ i, a i * A i ω| ≤ (∑ i, |a i|) * R := by |
| filter_upwards [ae_all_iff.mpr hA] with ω hω |
| calc |
| |∑ i, a i * A i ω| ≤ ∑ i, |a i * A i ω| := Finset.abs_sum_le_sum_abs _ _ |
| _ = ∑ i, |a i| * |A i ω| := by simp only [abs_mul] |
| _ ≤ ∑ i, |a i| * R := |
| Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hω i) (abs_nonneg _)) |
| _ = _ := (Finset.sum_mul _ _ _).symm |
| |
| omit [IsProbabilityMeasure μ] in |
| theorem martingale_sum {T : Type*} [Preorder T] |
| (F : Filtration T ‹MeasurableSpace Ω›) {M : ι → T → Ω → ℝ} |
| (hM : ∀ i, Martingale (M i) F μ) (a : ι → ℝ) : |
| Martingale (fun t ω => ∑ i, a i * M i t ω) F μ := by |
| classical |
| have hsum (s : Finset ι) : Martingale (fun t ω => ∑ i ∈ s, a i * M i t ω) F μ := by |
| induction s using Finset.induction_on with |
| | empty => |
| have heq : (fun t ω => ∑ i ∈ (∅ : Finset ι), a i * M i t ω) = |
| (0 : T → Ω → ℝ) := by funext t ω; simp |
| rw [heq] |
| exact martingale_zero ℝ F μ |
| | @insert i s hi ih => |
| have h := ((hM i).smul (a i)).add ih |
| have heq : a i • M i + (fun t ω => ∑ j ∈ s, a j * M j t ω) = |
| (fun t ω => ∑ j ∈ insert i s, a j * M j t ω) := by |
| funext t ω |
| simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Finset.sum_insert hi] |
| rw [heq] at h |
| exact h |
| exact hsum Finset.univ |
| |
| end Problem367.BrownianLinearCombination |
| |
| end |
| |
| end OAI |
| |
