Plain-language statement
The predictable part is additive for integrable processes.
Exact Lean statement
lemma predictablePart_add {ι Ω E : Type*} [Preorder ι] [LocallyFiniteOrderBot ι]
[SuccOrder ι] {mΩ : MeasurableSpace Ω} [NormedAddCommGroup E] [NormedSpace ℝ E]
[CompleteSpace E] {P : Measure Ω} {S₁ S₂ : ι → Ω → E} (𝓕 : Filtration ι mΩ)
(hS₁ : ∀ t, Integrable (S₁ t) P) (hS₂ : ∀ t, Integrable (S₂ t) P) (t : ι) :
predictablePart (S₁ + S₂) 𝓕 P t =ᵐ[P] predictablePart S₁ 𝓕 P t + predictablePart S₂ 𝓕 P tFormal artifact
Lean source
lemma predictablePart_add {ι Ω E : Type*} [Preorder ι] [LocallyFiniteOrderBot ι] [SuccOrder ι] {mΩ : MeasurableSpace Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {P : Measure Ω} {S₁ S₂ : ι → Ω → E} (𝓕 : Filtration ι mΩ) (hS₁ : ∀ t, Integrable (S₁ t) P) (hS₂ : ∀ t, Integrable (S₂ t) P) (t : ι) : predictablePart (S₁ + S₂) 𝓕 P t =ᵐ[P] predictablePart S₁ 𝓕 P t + predictablePart S₂ 𝓕 P t := by simp only [_root_.predictablePart, ← Finset.sum_add_distrib] refine eventuallyEq_sum fun i _ => ?_ rw [show (S₁ + S₂) (succ i) - (S₁ + S₂) i = (S₁ (succ i) - S₁ i) + (S₂ (succ i) - S₂ i) by simp; abel] exact condExp_add ((hS₁ (succ i)).sub (hS₁ i)) ((hS₂ (succ i)).sub (hS₂ i)) (𝓕 i)- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/DoobMeyer.lean:117-126
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Related declarations
Continuous Within At Iio indicator Ioc
continuousWithinAt_Iio_indicator_Ioc
Plain-language statement
The indicator of a half-open interval Ioc a b with constant value c is left-continuous: when approached from the left it is eventually constant, so it is continuous within Iio t at t for every t.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Iio ne Bot
Dense.comap_val_nhdsWithin_Iio_neBot
Plain-language statement
This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Ioi ne Bot
Dense.comap_val_nhdsWithin_Ioi_neBot
Project documentation
This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.
Source project: Brownian motion
Person-level attribution pending.