| import OAI.Probability.FreeIsing.Brownian.SimpleIntegral |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators Topology NNReal |
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| namespace Problem367.PredictableL2 |
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| theorem summable_of_orthogonal_sq |
| {E ι : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [CompleteSpace E] (f : ι → E) |
| (horth : Pairwise fun i j => inner ℝ (f i) (f j) = 0) |
| (hsum : Summable fun i => ‖f i‖ ^ 2) : Summable f := by |
| let V : ι → Submodule ℝ E := fun i => Submodule.span ℝ {f i} |
| have hV : OrthogonalFamily ℝ (fun i => V i) (fun i => (V i).subtypeₗᵢ) := by |
| apply OrthogonalFamily.of_pairwise |
| intro i j hij |
| change (V i) ⟂ (V j) |
| simpa only [V, Submodule.isOrtho_span, Set.mem_singleton_iff, |
| forall_eq] using horth hij |
| let x : (i : ι) → V i := fun i => ⟨f i, Submodule.mem_span_singleton_self (f i)⟩ |
| exact (hV.summable_iff_norm_sq_summable x).mpr hsum |
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| theorem norm_tsum_sq_of_orthogonal |
| {E ι : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [CompleteSpace E] (f : ι → E) |
| (horth : Pairwise fun i j => inner ℝ (f i) (f j) = 0) |
| (hsum : Summable fun i => ‖f i‖ ^ 2) : |
| ‖∑' i, f i‖ ^ 2 = ∑' i, ‖f i‖ ^ 2 := by |
| classical |
| have hf := summable_of_orthogonal_sq f horth hsum |
| have hfinite (s : Finset ι) : ‖∑ i ∈ s, f i‖ ^ 2 = ∑ i ∈ s, ‖f i‖ ^ 2 := by |
| rw [← real_inner_self_eq_norm_sq] |
| simp only [sum_inner, inner_sum] |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [Finset.sum_eq_single i] |
| · exact real_inner_self_eq_norm_sq (f i) |
| · intro j hj hji |
| exact horth hji |
| · exact fun h => (h hi).elim |
| have hn := (hf.hasSum.norm).pow 2 |
| simp_rw [hfinite] at hn |
| exact tendsto_nhds_unique hn hsum.hasSum |
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| theorem inner_toLp {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| {f g : Ω → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : |
| inner ℝ (hf.toLp f) (hg.toLp g) = ∫ ω, f ω * g ω ∂μ := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hf.coeFn_toLp, hg.coeFn_toLp] with ω hfω hgω |
| simp only [hfω, hgω, Real.inner_apply] |
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| theorem norm_toLp_sq {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| {f : Ω → ℝ} (hf : MemLp f 2 μ) : |
| ‖hf.toLp f‖ ^ 2 = ∫ ω, f ω ^ 2 ∂μ := by |
| rw [← real_inner_self_eq_norm_sq, inner_toLp] |
| simp only [sq] |
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| theorem toLp_finsetSum {Ω ι : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| (f : ι → Ω → ℝ) (hf : ∀ i, MemLp (f i) 2 μ) (s : Finset ι) : |
| (memLp_finsetSum s (fun i _ => hf i)).toLp (fun ω => ∑ i ∈ s, f i ω) = |
| ∑ i ∈ s, (hf i).toLp (f i) := by |
| apply Lp.ext |
| apply (MemLp.coeFn_toLp _).trans |
| apply Filter.EventuallyEq.symm |
| apply (Lp.coeFn_fun_finsetSum s (fun i => (hf i).toLp (f i))).trans |
| have heq : ∀ᵐ ω ∂μ, ∀ i ∈ s, (hf i).toLp (f i) ω = f i ω := |
| (Filter.eventually_all_finite s.finite_toSet).mpr (fun i _ => (hf i).coeFn_toLp) |
| filter_upwards [heq] with ω hω |
| exact Finset.sum_congr rfl (fun i hi => hω i hi) |
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| theorem exists_orthogonal_sum |
| {Ω ι : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| (f : ι → Ω → ℝ) (hf : ∀ i, MemLp (f i) 2 μ) |
| (horth : Pairwise fun i j => ∫ ω, f i ω * f j ω ∂μ = 0) |
| (henergy : Summable fun i => ∫ ω, f i ω ^ 2 ∂μ) : |
| ∃ (Y : Ω → ℝ) (hY : MemLp Y 2 μ), Measurable Y ∧ |
| HasSum (fun i => (hf i).toLp (f i)) (hY.toLp Y) ∧ |
| (∫ ω, Y ω ^ 2 ∂μ) = ∑' i, ∫ ω, f i ω ^ 2 ∂μ := by |
| let F : ι → Lp ℝ 2 μ := fun i => (hf i).toLp (f i) |
| have hForth : Pairwise fun i j => inner ℝ (F i) (F j) = 0 := by |
| intro i j hij |
| exact (inner_toLp (hf i) (hf j)).trans (horth hij) |
| have hFenergy : Summable fun i => ‖F i‖ ^ 2 := by |
| simpa only [F, norm_toLp_sq] using henergy |
| have hFsum := summable_of_orthogonal_sq F hForth hFenergy |
| let Y : Lp ℝ 2 μ := ∑' i, F i |
| refine ⟨Y, (Lp.memLp Y), (Lp.stronglyMeasurable Y).measurable, ?_, ?_⟩ |
| · simpa only [Lp.toLp_coeFn] using hFsum.hasSum |
| · rw [← norm_toLp_sq (Lp.memLp Y), Lp.toLp_coeFn] |
| exact (norm_tsum_sq_of_orthogonal F hForth hFenergy).trans |
| (by simp only [F, norm_toLp_sq]) |
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| theorem orthogonal_sum_unique |
| {Ω ι : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| (f : ι → Ω → ℝ) (hf : ∀ i, MemLp (f i) 2 μ) |
| {Y Z : Ω → ℝ} (hY : MemLp Y 2 μ) (hZ : MemLp Z 2 μ) |
| (hYsum : HasSum (fun i => (hf i).toLp (f i)) (hY.toLp Y)) |
| (hZsum : HasSum (fun i => (hf i).toLp (f i)) (hZ.toLp Z)) : |
| Y =ᵐ[μ] Z := |
| (hY.toLp_eq_toLp_iff hZ).mp (hYsum.unique hZsum) |
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| theorem tendsto_integral_sq_error_of_hasSum |
| {Ω ι : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| (f : ι → Ω → ℝ) (hf : ∀ i, MemLp (f i) 2 μ) |
| {Y : Ω → ℝ} (hY : MemLp Y 2 μ) |
| (hYsum : HasSum (fun i => (hf i).toLp (f i)) (hY.toLp Y)) : |
| Tendsto (fun s : Finset ι => ∫ ω, ((∑ i ∈ s, f i ω) - Y ω) ^ 2 ∂μ) |
| atTop (𝓝 0) := by |
| have h := ((hYsum.sub_const (hY.toLp Y)).norm).pow 2 |
| change Tendsto _ atTop _ at h |
| simp only [sub_self, norm_zero, zero_pow (by decide : 2 ≠ 0)] at h |
| convert h using 1 |
| funext s |
| calc |
| _ = ‖((memLp_finsetSum s (fun i _ => hf i)).sub hY).toLp |
| ((fun ω => ∑ i ∈ s, f i ω) - Y)‖ ^ 2 := |
| (norm_toLp_sq ((memLp_finsetSum s (fun i _ => hf i)).sub hY)).symm |
| _ = _ := by |
| change ‖(memLp_finsetSum s (fun i _ => hf i)).toLp |
| (fun ω => ∑ i ∈ s, f i ω) - hY.toLp Y‖ ^ 2 = _ |
| rw [toLp_finsetSum] |
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| theorem hasSum_integral_mul_of_hasSum |
| {Ω ι : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| (f : ι → Ω → ℝ) (hf : ∀ i, MemLp (f i) 2 μ) |
| {Y Z : Ω → ℝ} (hY : MemLp Y 2 μ) (hZ : MemLp Z 2 μ) |
| (hYsum : HasSum (fun i => (hf i).toLp (f i)) (hY.toLp Y)) : |
| HasSum (fun i => ∫ ω, Z ω * f i ω ∂μ) (∫ ω, Z ω * Y ω ∂μ) := by |
| have h := hYsum.mapL (innerSL ℝ (hZ.toLp Z)) |
| change HasSum (fun i => inner ℝ (hZ.toLp Z) ((hf i).toLp (f i))) |
| (inner ℝ (hZ.toLp Z) (hY.toLp Y)) at h |
| simpa only [inner_toLp] using h |
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| theorem integral_eq_zero_of_hasSum |
| {Ω ι : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] |
| (f : ι → Ω → ℝ) (hf : ∀ i, MemLp (f i) 2 μ) |
| {Y : Ω → ℝ} (hY : MemLp Y 2 μ) |
| (hYsum : HasSum (fun i => (hf i).toLp (f i)) (hY.toLp Y)) |
| (hcenter : ∀ i, ∫ ω, f i ω ∂μ = 0) : ∫ ω, Y ω ∂μ = 0 := by |
| have h := hasSum_integral_mul_of_hasSum f hf hY |
| (memLp_const (1 : ℝ)) hYsum |
| simp only [one_mul, hcenter] at h |
| exact h.unique hasSum_zero |
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| section Predictable |
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| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] |
| (ℱ : Filtration ℕ mΩ) (H D : ℕ → Ω → ℝ) (v : ℕ → ℝ) |
| (hH : ∀ i, StronglyMeasurable[ℱ i] (H i)) |
| (hD : ∀ i, StronglyMeasurable[ℱ (i + 1)] (D i)) |
| (hD2 : ∀ i, MemLp (D i) 2 μ) |
| (hHD : ∀ i, MemLp (fun ω => H i ω * D i ω) 2 μ) |
| (hzero : ∀ i, μ[D i | ℱ i] =ᵐ[μ] 0) |
| (hvar : ∀ i, μ[(fun ω => D i ω ^ 2) | ℱ i] =ᵐ[μ] fun _ => v i) |
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| include hH hD hD2 hzero hvar in |
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| theorem exists_predictable_sum |
| (henergy : Summable fun i => v i * ∫ ω, H i ω ^ 2 ∂μ) : |
| ∃ (Y : Ω → ℝ) (hY : MemLp Y 2 μ), Measurable Y ∧ |
| HasSum (fun i => (hHD i).toLp (fun ω => H i ω * D i ω)) (hY.toLp Y) ∧ |
| (∫ ω, Y ω ^ 2 ∂μ) = ∑' i, v i * ∫ ω, H i ω ^ 2 ∂μ := by |
| have horth : Pairwise fun i j => |
| ∫ ω, (H i ω * D i ω) * (H j ω * D j ω) ∂μ = 0 := by |
| intro i j hij |
| rcases lt_or_gt_of_ne hij with hij | hji |
| · exact SimpleIntegral.predictable_increments_orthogonal ℱ H D hH hD |
| (fun i => (hD2 i).integrable (by norm_num)) hHD hzero hij |
| · simpa only [mul_comm] using |
| SimpleIntegral.predictable_increments_orthogonal ℱ H D hH hD |
| (fun i => (hD2 i).integrable (by norm_num)) hHD hzero hji |
| have hsecond (i : ℕ) : |
| (∫ ω, (H i ω * D i ω) ^ 2 ∂μ) = v i * ∫ ω, H i ω ^ 2 ∂μ := |
| SimpleIntegral.integral_sq_mul_of_condExp_sq (ℱ.le i) (hH i) |
| (hD2 i).integrable_sq (hHD i).integrable_sq (hvar i) |
| obtain ⟨Y, hY, hmY, hsY, hiY⟩ := exists_orthogonal_sum |
| (fun i ω => H i ω * D i ω) hHD horth (by simpa only [hsecond] using henergy) |
| exact ⟨Y, hY, hmY, hsY, by simpa only [hsecond] using hiY⟩ |
| |
| end Predictable |
| |
| theorem exists_preBrownian_predictable_sum |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P) |
| (hm : ∀ t, StronglyMeasurable (B t)) |
| (t : ℕ → ℝ≥0) (ht : Monotone t) (H : ℕ → Ω → ℝ) |
| (hH : ∀ i, StronglyMeasurable[Filtration.natural B hm (t i)] (H i)) |
| (hH2 : ∀ i, MemLp (H i) 2 P) |
| (henergy : Summable fun i => |
| ((t (i + 1) : ℝ) - (t i : ℝ)) * ∫ ω, H i ω ^ 2 ∂P) : |
| ∃ Y : Ω → ℝ, Measurable Y ∧ MemLp Y 2 P ∧ |
| (∫ ω, Y ω ∂P) = 0 ∧ |
| (∫ ω, Y ω ^ 2 ∂P) = |
| ∑' i, ((t (i + 1) : ℝ) - (t i : ℝ)) * ∫ ω, H i ω ^ 2 ∂P ∧ |
| Tendsto (fun s : Finset ℕ => ∫ ω, |
| ((∑ i ∈ s, H i ω * (B (t (i + 1)) ω - B (t i) ω)) - Y ω) ^ 2 ∂P) |
| atTop (𝓝 0) := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| let ℱ : Filtration ℕ mΩ := |
| ⟨fun i => Filtration.natural B hm (t i), |
| fun _ _ hij => (Filtration.natural B hm).mono (ht hij), |
| fun i => (Filtration.natural B hm).le (t i)⟩ |
| let D : ℕ → Ω → ℝ := fun i ω => B (t (i + 1)) ω - B (t i) ω |
| have hD : ∀ i, StronglyMeasurable[ℱ (i + 1)] (D i) := by |
| intro i |
| exact (Filtration.stronglyAdapted_natural hm (t (i + 1))).sub |
| ((Filtration.stronglyAdapted_natural hm (t i)).mono |
| ((Filtration.natural B hm).mono (ht (Nat.le_succ i)))) |
| have hD2 : ∀ i, MemLp (D i) 2 P := by |
| intro i |
| exact (hB.isGaussianProcess.hasGaussianLaw_eval (t (i + 1))).memLp_two.sub |
| (hB.isGaussianProcess.hasGaussianLaw_eval (t i)).memLp_two |
| have hHD : ∀ i, MemLp (fun ω => H i ω * D i ω) 2 P := fun i => |
| preBrownian_predictable_mul_memLp_two hB hm (ht (Nat.le_succ i)) (hH i) (hH2 i) |
| have hz : ∀ i, P[D i | ℱ i] =ᵐ[P] 0 := fun i => |
| preBrownian_increment_condExp_zero hB hm (ht (Nat.le_succ i)) |
| obtain ⟨Y, hY, hmY, hsY, hiY⟩ := exists_predictable_sum ℱ H D |
| (fun i => (t (i + 1) : ℝ) - (t i : ℝ)) hH hD hD2 hHD hz |
| (fun i => preBrownian_increment_sq_condExp hB hm (ht (Nat.le_succ i))) henergy |
| refine ⟨Y, hmY, hY, ?_, hiY, ?_⟩ |
| · apply integral_eq_zero_of_hasSum (fun i ω => H i ω * D i ω) hHD hY hsY |
| intro i |
| exact SimpleIntegral.integral_mul_eq_zero_of_condExp_eq_zero (ℱ.le i) (hH i) |
| ((hD2 i).integrable (by norm_num)) ((hHD i).integrable (by norm_num)) (hz i) |
| · exact tendsto_integral_sq_error_of_hasSum (fun i ω => H i ω * D i ω) hHD hY hsY |
|
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| end Problem367.PredictableL2 |
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| end |
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| end OAI |
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