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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
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