File size: 3,483 Bytes
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 | import OAI.Probability.FreeIsing.Brownian.SpatialAssociativity
import OAI.Probability.FreeIsing.Stochastic.StoppedIntegralAssociativity
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.StoppedIntegralLocality
variable {Ω : Type*}
theorem uniformSum_stopped_congr (M H K : ℝ≥0 → Ω → ℝ)
(τ : Ω → WithTop ℝ≥0) (T : ℝ≥0) (n : ℕ) (ω : Ω)
(hlocal : ∀ t : ℝ≥0, (t : WithTop ℝ≥0) ≤ τ ω → H t ω = K t ω) :
BrownianGrid.uniformSum (stoppedProcess M τ) H T n ω =
BrownianGrid.uniformSum (stoppedProcess M τ) K T n ω := by
unfold BrownianGrid.uniformSum BrownianGrid.integralSum
apply Finset.sum_congr rfl
intro i hi
dsimp only
by_cases ht : (BrownianGrid.uniformGrid T n i : WithTop ℝ≥0) ≤ τ ω
· rw [hlocal _ ht]
· have ha : τ ω ≤ (BrownianGrid.uniformGrid T n i : WithTop ℝ≥0) :=
le_of_not_ge ht
have hab : (BrownianGrid.uniformGrid T n i : WithTop ℝ≥0) ≤
(BrownianGrid.uniformGrid T n (i + 1) : WithTop ℝ≥0) := by
exact_mod_cast BrownianGrid.uniformGrid_mono T n (Nat.le_succ i)
rw [stoppedProcess_eq_of_ge ha, stoppedProcess_eq_of_ge (ha.trans hab)]
simp only [sub_self, mul_zero]
variable [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
theorem stopped_integral_eq_of_local_eq
{F : Filtration ℝ≥0 mΩ} {M H K I J : ℝ≥0 → Ω → ℝ}
(hM : Martingale M F P) (hM2 : ∀ t, MemLp (M t) 2 P)
(hcM : ∀ᵐ ω ∂P, Continuous (fun t => M t ω))
(hH : StronglyAdapted F H) (hK : StronglyAdapted F K)
{C D : ℝ} (hbH : ∀ᵐ ω ∂P, ∀ t, |H t ω| ≤ C)
(hbK : ∀ᵐ ω ∂P, ∀ t, |K t ω| ≤ D)
(hI : Martingale I F P) (hJ : Martingale J F P)
(hI2 : ∀ t, MemLp (I t) 2 P) (hJ2 : ∀ t, MemLp (J t) 2 P)
(hcI : ∀ᵐ ω ∂P, Continuous (fun t => I t ω))
(hcJ : ∀ᵐ ω ∂P, Continuous (fun t => J t ω))
(T : ℝ≥0)
(hIlim : Tendsto (fun n => ∫ ω,
(BrownianGrid.uniformSum M H T n ω - I T ω) ^ 2 ∂P) atTop (𝓝 0))
(hJlim : Tendsto (fun n => ∫ ω,
(BrownianGrid.uniformSum M K T n ω - J T ω) ^ 2 ∂P) atTop (𝓝 0))
{τ : Ω → WithTop ℝ≥0} (hτ : IsStoppingTime F τ)
(hlocal : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, (t : WithTop ℝ≥0) ≤ τ ω → H t ω = K t ω) :
stoppedProcess I τ T =ᵐ[P] stoppedProcess J τ T := by
have hIs := StoppedMartingaleL2.memLp_stoppedProcess_two hI hI2 hcI hτ T
have hJs := StoppedMartingaleL2.memLp_stoppedProcess_two hJ hJ2 hcJ hτ T
have hMs (t : ℝ≥0) :=
StoppedMartingaleL2.memLp_stoppedProcess_two hM hM2 hcM hτ t
have hX (n : ℕ) : MemLp (BrownianGrid.uniformSum (stoppedProcess M τ) H T n) 2 P :=
BrownianSpatialAssociativity.uniformSum_memLp hMs
(fun t => ((hH t).mono (F.le t)).aestronglyMeasurable)
(fun t => hbH.mono (fun ω hω => hω t)) T n
have hi := StoppedIntegralAssociativity.tendsto_stopped_outer_uniformSum
hM hM2 hcM hH hbH hI hI2 hcI T hIlim hτ
have hj := StoppedIntegralAssociativity.tendsto_stopped_outer_uniformSum
hM hM2 hcM hK hbK hJ hJ2 hcJ T hJlim hτ
apply BrownianGrid.ae_eq_of_mean_square_limits hX hIs hJs hi
apply hj.congr
intro n
apply integral_congr_ae
filter_upwards [hlocal] with ω hω
rw [uniformSum_stopped_congr M H K τ T n ω hω]
end Problem367.StoppedIntegralLocality
end
end OAI
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