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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import OAI.Probability.FreeIsing.Stochastic.StoppedBrownianCross
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.StoppedBrownianArraySquare
open StoppedSimpleIntegral BrownianFiltration StoppedBrownianSquare StoppedBrownianCross
variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
{F : Filtration ℝ≥0 mΩ}
private theorem finite_sum_martingale [IsFiniteMeasure P]
(S : ι → ℝ≥0 → Ω → ℝ) (I : Finset ι)
(hS : ∀ i ∈ I, Martingale (S i) F P) :
Martingale (fun t ω => ∑ i ∈ I, S i t ω) F P := by
classical
induction I using Finset.induction_on with
| empty =>
convert martingale_zero ℝ F P using 1
funext t ω
simp
| @insert i I hi ih =>
have hm := (hS i (Finset.mem_insert_self i I)).add
(ih (fun j hj => hS j (Finset.mem_insert_of_mem hj)))
convert hm using 1
funext t ω
simp only [Finset.sum_insert hi, Pi.add_apply]
theorem finite_sum_compensated_square_martingale [IsFiniteMeasure P]
(S A : ι → ℝ≥0 → Ω → ℝ)
(hdiag : ∀ i, Martingale (fun t ω => S i t ω ^ 2 - A i t ω) F P)
(hcross : Pairwise fun i j => Martingale (fun t ω => S i t ω * S j t ω) F P)
(I : Finset ι) :
Martingale (fun t ω => (∑ i ∈ I, S i t ω) ^ 2 - ∑ i ∈ I, A i t ω) F P := by
classical
induction I using Finset.induction_on with
| empty =>
convert martingale_zero ℝ F P using 1
funext t ω
simp
| @insert i I hi ih =>
have hm := finite_sum_martingale (fun j t ω => S i t ω * S j t ω) I
(fun j hj => hcross (fun heq => hi (heq ▸ hj)))
convert ((hdiag i).add ih).add (Martingale.smul 2 hm) using 1
funext t ω
simp only [Finset.sum_insert hi, Pi.add_apply, Pi.smul_apply, smul_eq_mul,
← Finset.mul_sum]
ring
theorem stopped_brownian_array_square_martingale [IsProbabilityMeasure P]
(B : ι → ℝ≥0 → Ω → ℝ)
(hB : ∀ j, IsPreBrownianReal (B j) P)
(hAd : ∀ j, StronglyAdapted F (B j))
(hpast : ∀ s t, s ≤ t → Indep (MeasurableSpace.comap
(fun ω j => B j t ω - B j s ω) inferInstance) (F s) P)
(hind : ∀ s t, s ≤ t → Pairwise fun j k =>
IndepFun (fun ω => B j t ω - B j s ω) (fun ω => B k t ω - B k s ω) P)
(τ : ℕ → ℝ≥0) (hτ : Monotone τ) (H : ι → ℕ → Ω → ℝ)
(hH : ∀ j i, StronglyMeasurable[F (τ i)] (H j i))
(K : ι → ℕ → ℝ) (hbound : ∀ j i, ∀ᵐ ω ∂P, ‖H j i ω‖ ≤ K j i)
(I : Finset ι) (n : ℕ) :
Martingale (fun t ω =>
(∑ j ∈ I, ∑ i ∈ Finset.range n,
intervalSum (B j) (H j i) (τ i) (τ (i + 1)) t ω) ^ 2 -
∑ j ∈ I, ∑ i ∈ Finset.range n,
H j i ω ^ 2 * ((min t (τ (i + 1)) : ℝ) - (min t (τ i) : ℝ))) F P := by
have hcoord (j : ι) (s t : ℝ≥0) (hst : s ≤ t) : Indep (MeasurableSpace.comap
(fun ω => B j t ω - B j s ω) inferInstance) (F s) P := by
apply indep_of_indep_of_le_left (hpast s t hst)
have hm : Measurable[MeasurableSpace.comap
(fun ω j => B j t ω - B j s ω) inferInstance]
(fun ω j => B j t ω - B j s ω) := measurable_iff_comap_le.mpr le_rfl
exact ((measurable_pi_apply j).comp hm).comap_le
have hpair (j k : ι) (s t : ℝ≥0) (hst : s ≤ t) : Indep (MeasurableSpace.comap
(fun ω => (B j t ω - B j s ω, B k t ω - B k s ω)) inferInstance) (F s) P := by
apply indep_of_indep_of_le_left (hpast s t hst)
have hm : Measurable[MeasurableSpace.comap
(fun ω j => B j t ω - B j s ω) inferInstance]
(fun ω j => B j t ω - B j s ω) := measurable_iff_comap_le.mpr le_rfl
exact (((measurable_pi_apply j).comp hm).prodMk
((measurable_pi_apply k).comp hm)).comap_le
exact finite_sum_compensated_square_martingale
(fun j t ω => ∑ i ∈ Finset.range n,
intervalSum (B j) (H j i) (τ i) (τ (i + 1)) t ω)
(fun j t ω => ∑ i ∈ Finset.range n,
H j i ω ^ 2 * ((min t (τ (i + 1)) : ℝ) - (min t (τ i) : ℝ)))
(fun j => stopped_predictable_sum_square_martingale (hB j) (hAd j) (hcoord j)
τ hτ (H j) (hH j) (K j) (hbound j) n)
(fun j k hjk => stopped_brownian_sum_product_martingale (hB j) (hB k)
(hAd j) (hAd k) (hpair j k) (fun s t hst => hind s t hst hjk)
τ hτ (H j) (H k) (hH j) (hH k) (K j) (K k) (hbound j) (hbound k) n) I
end Problem367.StoppedBrownianArraySquare
end
end OAI
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