File size: 2,493 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 | import OAI.Probability.FreeIsing.Estimates.CompensatedSquareLimit
import OAI.Probability.FreeIsing.Estimates.NonnegativeSeriesL1
import OAI.Probability.FreeIsing.Martingales.PredictableL2Completion
import OAI.Probability.FreeIsing.Stochastic.StoppedBrownianArraySquare
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology ENNReal NNReal
namespace Problem367.MartingaleSpatialCompensator
variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω]
{μ : Measure Ω} [IsFiniteMeasure μ] {F : Filtration ℝ≥0 mΩ}
theorem spatial_sum_compensatedSquare_martingale [Countable ι]
{X A : ι → ℝ≥0 → Ω → ℝ} {Y : ℝ≥0 → Ω → ℝ}
(hX2 : ∀ i t, MemLp (X i t) 2 μ)
(hdiag : ∀ i, Martingale (fun t ω => X i t ω ^ 2 - A i t ω) F μ)
(hcross : Pairwise fun i j => Martingale (fun t ω => X i t ω * X j t ω) F μ)
(hAad : ∀ i, StronglyAdapted F (A i))
(hAint : ∀ i t, Integrable (A i t) μ)
(hAnonneg : ∀ i t ω, 0 ≤ A i t ω)
(hAsum : ∀ t, Summable (fun i => ∫ ω, A i t ω ∂μ))
(hYad : StronglyAdapted F Y) (hY2 : ∀ t, MemLp (Y t) 2 μ)
(hYsum : ∀ t, HasSum (fun i => (hX2 i t).toLp (X i t)) ((hY2 t).toLp (Y t))) :
Martingale (fun t ω => Y t ω ^ 2 - ∑' i, A i t ω) F μ := by
classical
apply CompensatedSquareLimit.martingale_compensatedSquare_of_limits
(l := (atTop : Filter (Finset ι)))
(X := fun s t ω => ∑ i ∈ s, X i t ω)
(A := fun s t ω => ∑ i ∈ s, A i t ω)
· intro s t
exact memLp_finsetSum s (fun i _ => hX2 i t)
· intro s t
exact integrable_finsetSum s (fun i _ => hAint i t)
· exact StoppedBrownianArraySquare.finite_sum_compensated_square_martingale X A hdiag hcross
· exact hYad
· exact hY2
· intro t
let : MeasurableSpace Ω := F t
exact StronglyMeasurable.tsum (fun i => hAad i t)
· intro t
exact (NonnegativeSeriesL1.integrable_tsum_of_nonnegative
(fun i => hAint i t) (fun i ω => hAnonneg i t ω) (hAsum t)).2.1
· intro t
apply CompensatedSquareLimit.tendsto_eLpNorm_two_of_meanSquare
(fun s => memLp_finsetSum s (fun i _ => hX2 i t)) (hY2 t)
exact PredictableL2.tendsto_integral_sq_error_of_hasSum
(fun i => X i t) (fun i => hX2 i t) (hY2 t) (hYsum t)
· intro t
exact NonnegativeSeriesL1.tendsto_finsetSum_L1_of_nonnegative
(fun i => hAint i t) (fun i ω => hAnonneg i t ω) (hAsum t)
end Problem367.MartingaleSpatialCompensator
end
end OAI
|