| import Mathlib | |
| namespace OAI | |
| open MeasureTheory | |
| namespace Problem367.SimpleIntegral | |
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} | |
| theorem integral_mul_eq_zero_of_condExp_eq_zero [IsFiniteMeasure μ] | |
| {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {H D : Ω → ℝ} | |
| (hH : StronglyMeasurable[m] H) (hD : Integrable D μ) | |
| (hHD : Integrable (H * D) μ) (hzero : μ[D | m] =ᵐ[μ] 0) : | |
| ∫ ω, H ω * D ω ∂μ = 0 := by | |
| have hpull := condExp_mul_of_stronglyMeasurable_left hH hHD hD | |
| have hCE : μ[H * D | m] =ᵐ[μ] 0 := by | |
| filter_upwards [hpull, hzero] with ω hp hz | |
| simp only [Pi.mul_apply, Pi.zero_apply] at hp hz ⊢ | |
| rw [hp, hz, mul_zero] | |
| calc | |
| ∫ ω, H ω * D ω ∂μ = ∫ ω, μ[H * D | m] ω ∂μ := | |
| (integral_condExp hm).symm | |
| _ = 0 := by rw [integral_congr_ae hCE]; simp | |
| theorem integral_sq_mul_of_condExp_sq [IsFiniteMeasure μ] | |
| {m : MeasurableSpace Ω} (hm : m ≤ mΩ) {H D : Ω → ℝ} {v : ℝ} | |
| (hH : StronglyMeasurable[m] H) | |
| (hDsq : Integrable (fun ω => D ω ^ 2) μ) | |
| (hHDsq : Integrable (fun ω => (H ω * D ω) ^ 2) μ) | |
| (hvar : μ[(fun ω => D ω ^ 2) | m] =ᵐ[μ] fun _ => v) : | |
| ∫ ω, (H ω * D ω) ^ 2 ∂μ = v * ∫ ω, H ω ^ 2 ∂μ := by | |
| have hHsq : StronglyMeasurable[m] (fun ω => H ω ^ 2) := by | |
| simpa only [pow_two, Pi.mul_def] using hH.mul hH | |
| have hprod : Integrable ((fun ω => H ω ^ 2) * (fun ω => D ω ^ 2)) μ := by | |
| simpa only [Pi.mul_def, mul_pow] using hHDsq | |
| have hpull := condExp_mul_of_stronglyMeasurable_left hHsq hprod hDsq | |
| calc | |
| ∫ ω, (H ω * D ω) ^ 2 ∂μ = | |
| ∫ ω, μ[(fun ω => H ω ^ 2) * (fun ω => D ω ^ 2) | m] ω ∂μ := by | |
| simpa only [Pi.mul_def, mul_pow] using (integral_condExp hm).symm | |
| _ = ∫ ω, v * H ω ^ 2 ∂μ := by | |
| apply integral_congr_ae | |
| filter_upwards [hpull, hvar] with ω hp hv | |
| simp only [Pi.mul_apply] at hp | |
| rw [hp, hv, mul_comm] | |
| _ = v * ∫ ω, H ω ^ 2 ∂μ := integral_const_mul _ _ | |
| theorem integral_sq_sum_of_orthogonal {ι : Type*} (s : Finset ι) | |
| (Y : ι → Ω → ℝ) | |
| (hint : ∀ i ∈ s, ∀ j ∈ s, Integrable (fun ω => Y i ω * Y j ω) μ) | |
| (horth : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → ∫ ω, Y i ω * Y j ω ∂μ = 0) : | |
| ∫ ω, (∑ i ∈ s, Y i ω) ^ 2 ∂μ = ∑ i ∈ s, ∫ ω, Y i ω ^ 2 ∂μ := by | |
| classical | |
| calc | |
| ∫ ω, (∑ i ∈ s, Y i ω) ^ 2 ∂μ = | |
| ∫ ω, ∑ i ∈ s, ∑ j ∈ s, Y i ω * Y j ω ∂μ := by | |
| congr 1 | |
| funext ω | |
| simp only [pow_two, Finset.sum_mul_sum] | |
| _ = ∑ i ∈ s, ∑ j ∈ s, ∫ ω, Y i ω * Y j ω ∂μ := by | |
| rw [integral_finsetSum s (fun i hi => | |
| integrable_finsetSum s (fun j hj => hint i hi j hj))] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| exact integral_finsetSum s (fun j hj => hint i hi j hj) | |
| _ = ∑ i ∈ s, ∫ ω, Y i ω ^ 2 ∂μ := by | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| rw [Finset.sum_eq_single i] | |
| · simp only [pow_two] | |
| · intro j hj hji | |
| exact horth i hi j hj hji.symm | |
| · exact fun h => (h hi).elim | |
| theorem predictable_increments_orthogonal [IsFiniteMeasure μ] | |
| (ℱ : Filtration ℕ mΩ) (H D : ℕ → Ω → ℝ) | |
| (hH : ∀ i, StronglyMeasurable[ℱ i] (H i)) | |
| (hD : ∀ i, StronglyMeasurable[ℱ (i + 1)] (D i)) | |
| (hDint : ∀ i, Integrable (D i) μ) | |
| (hHD : ∀ i, MemLp (fun ω => H i ω * D i ω) 2 μ) | |
| (hzero : ∀ i, μ[D i | ℱ i] =ᵐ[μ] 0) | |
| {i j : ℕ} (hij : i < j) : | |
| ∫ ω, (H i ω * D i ω) * (H j ω * D j ω) ∂μ = 0 := by | |
| have hcoeff : StronglyMeasurable[ℱ j] | |
| (fun ω => (H i ω * D i ω) * H j ω) := by | |
| simpa only [Pi.mul_def] using | |
| (((hH i).mono (ℱ.mono hij.le)).mul | |
| ((hD i).mono (ℱ.mono (Nat.succ_le_of_lt hij)))).mul (hH j) | |
| have hprod : Integrable | |
| ((fun ω => (H i ω * D i ω) * H j ω) * D j) μ := by | |
| simpa only [Pi.mul_def, mul_assoc] using (hHD i).integrable_mul (hHD j) | |
| simpa only [mul_assoc] using | |
| integral_mul_eq_zero_of_condExp_eq_zero (ℱ.le j) hcoeff (hDint j) hprod (hzero j) | |
| theorem predictable_sum_isometry [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) | |
| (s : Finset ℕ) : | |
| ∫ ω, (∑ i ∈ s, H i ω * D i ω) ^ 2 ∂μ = | |
| ∑ i ∈ s, v i * ∫ ω, H i ω ^ 2 ∂μ := by | |
| have hint : ∀ i ∈ s, ∀ j ∈ s, | |
| Integrable (fun ω => (H i ω * D i ω) * (H j ω * D j ω)) μ := by | |
| intro i _ j _ | |
| simpa only [Pi.mul_def] using (hHD i).integrable_mul (hHD j) | |
| have hDint : ∀ i, Integrable (D i) μ := fun i => | |
| MemLp.integrable (by norm_num) (hD2 i) | |
| have horth : ∀ i ∈ s, ∀ j ∈ s, 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 predictable_increments_orthogonal ℱ H D hH hD hDint hHD hzero hij | |
| · simpa only [mul_comm] using | |
| predictable_increments_orthogonal ℱ H D hH hD hDint hHD hzero hji | |
| rw [integral_sq_sum_of_orthogonal s (fun i ω => H i ω * D i ω) hint horth] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| exact integral_sq_mul_of_condExp_sq (ℱ.le i) (hH i) | |
| (hD2 i).integrable_sq (hHD i).integrable_sq (hvar i) | |
| end Problem367.SimpleIntegral | |
| end OAI | |