| import OAI.Probability.FreeIsing.Brownian.FiltrationIntegral |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
|
|
| namespace Problem367.BrownianFiltration |
|
|
| theorem square_sub_time_martingale |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} |
| {B : ℝ≥0 → Ω → ℝ} {F : Filtration ℝ≥0 mΩ} |
| (hB : IsPreBrownianReal B P) (hAd : StronglyAdapted F B) |
| (hind : ∀ s t, s ≤ t → Indep |
| (MeasurableSpace.comap (fun ω => B t ω - B s ω) inferInstance) (F s) P) : |
| Martingale (fun t ω => B t ω ^ 2 - (t : ℝ)) F P := by |
| let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure |
| have hL2 (t) := (hB.isGaussianProcess.hasGaussianLaw_eval t).memLp_two |
| refine ⟨fun t => (hAd t).pow 2 |>.sub stronglyMeasurable_const, ?_⟩ |
| intro s t hst |
| let G := F s |
| let X : Ω → ℝ := B t - B s |
| have hXL2 : MemLp X 2 P := (hL2 t).sub (hL2 s) |
| have hXint : Integrable X P := (hB.integrable_eval t).sub (hB.integrable_eval s) |
| have hXzero : P[X | G] =ᵐ[P] fun _ => 0 := |
| increment_condExp_zero (mΩ := mΩ) hB hAd (hind s t hst) |
| have hprodInt : Integrable (B s * X) P := (hL2 s).integrable_mul hXL2 |
| have hprodZero : P[B s * X | G] =ᵐ[P] fun _ => 0 := by |
| have hp := condExp_mul_of_stronglyMeasurable_left (hAd s) hprodInt hXint |
| filter_upwards [hp, hXzero] with ω hp hz |
| change P[B s * X | G] ω = B s ω * P[X | G] ω at hp |
| rw [hz, mul_zero] at hp |
| exact hp |
| have htwoZero : P[(fun ω => 2 * (B s ω * X ω)) | G] =ᵐ[P] fun _ => 0 := by |
| have hp := condExp_smul (μ := P) (2 : ℝ) (B s * X) G |
| filter_upwards [hp, hprodZero] with ω hp hz |
| change P[(fun ω => 2 * (B s ω * X ω)) | G] ω = 2 * P[B s * X | G] ω at hp |
| rw [hz, mul_zero] at hp |
| exact hp |
| have hXSq := increment_sq_condExp (mΩ := mΩ) hB hAd hst (hind s t hst) |
| have hBSq : P[(fun ω => (B s ω)^2) | G] = (fun ω => (B s ω)^2) := |
| condExp_of_stronglyMeasurable (F.le s) |
| ((hAd s).pow 2) (hL2 s).integrable_sq |
| have hfun : (fun ω => (B t ω)^2) = |
| (fun ω => (X ω)^2 + 2 * (B s ω * X ω)) + (fun ω => (B s ω)^2) := by |
| funext ω |
| dsimp [X] |
| ring |
| have hsum1 := condExp_add hXL2.integrable_sq (hprodInt.const_mul 2) G |
| have hsum2 := condExp_add (hXL2.integrable_sq.add (hprodInt.const_mul 2)) |
| (hL2 s).integrable_sq G |
| have hBtSq : P[(fun ω => (B t ω)^2) | G] =ᵐ[P] |
| fun ω => ((t : ℝ) - (s : ℝ)) + (B s ω)^2 := by |
| rw [hfun] |
| filter_upwards [hsum2, hsum1, htwoZero, hXSq] with ω h2 h1 hz hsq |
| change P[(fun ω => (X ω)^2) | G] ω = (t : ℝ) - (s : ℝ) at hsq |
| simp only [Pi.add_def, Pi.mul_def] at h2 h1 |
| rw [hBSq] at h2 |
| rw [h1, hz, hsq] at h2 |
| simpa only [Pi.add_def, add_zero] using h2 |
| have hsub := condExp_sub (hL2 t).integrable_sq (integrable_const (t : ℝ)) G |
| have hc : P[(fun _ : Ω => (t : ℝ)) | G] = (fun _ => (t : ℝ)) := |
| condExp_const (F.le s) (t : ℝ) |
| filter_upwards [hsub, hBtSq] with ω hs hsq |
| rw [hc] at hs |
| simp only [Pi.sub_apply] at hs |
| rw [hsq] at hs |
| simp only [Pi.sub_def] at hs |
| exact hs.trans (by ring) |
|
|
| end Problem367.BrownianFiltration |
|
|
| end |
|
|
| end OAI |
|
|
