Predictable Seq Step apply
predictableSeqStep_apply
Plain-language statement
On the mesh cell Ioc (pred u) u containing t, the step process predictableSeqStep is constant, equal to the discrete predictable part at the cell's right endpoint u.
Exact Lean statement
lemma predictableSeqStep_apply {ι Ω : Type*} [TopologicalSpace ι] [SecondCountableTopology ι]
[LinearOrder ι] [OrderBot ι] [OrderTop ι] {mΩ : MeasurableSpace Ω} (P : Measure Ω)
(S : ι → Ω → ℝ) (𝓕 : Filtration ι mΩ) (n : ℕ) (ω : Ω) {t : ι} {u : mesh ι n}
(ht : t ∈ meshPredIoc n u) :
predictableSeqStep P S 𝓕 n t ω
= predictablePart (S ∘ Subtype.val) (meshFiltration 𝓕 n) P u ωFormal artifact
Lean source
lemma predictableSeqStep_apply {ι Ω : Type*} [TopologicalSpace ι] [SecondCountableTopology ι] [LinearOrder ι] [OrderBot ι] [OrderTop ι] {mΩ : MeasurableSpace Ω} (P : Measure Ω) (S : ι → Ω → ℝ) (𝓕 : Filtration ι mΩ) (n : ℕ) (ω : Ω) {t : ι} {u : mesh ι n} (ht : t ∈ meshPredIoc n u) : predictableSeqStep P S 𝓕 n t ω = predictablePart (S ∘ Subtype.val) (meshFiltration 𝓕 n) P u ω := by rw [predictableSeqStep_eq_sum_indicator, Finset.sum_eq_single_of_mem u (Finset.mem_univ _) ?_, Set.indicator_of_mem ht] refine fun v _ hvu => Set.indicator_of_notMem (fun hv => hvu ?_) _ rcases lt_trichotomy v u with h' | h' | h' · exact absurd (lt_of_le_of_lt hv.2 (lt_of_le_of_lt (Subtype.coe_le_coe.2 (Order.le_pred_of_lt h')) ht.1)) (lt_irrefl t) · exact h' · exact absurd (lt_of_le_of_lt ht.2 (lt_of_le_of_lt (Subtype.coe_le_coe.2 (Order.le_pred_of_lt h')) hv.1)) (lt_irrefl t)- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/DoobMeyer.lean:1048-1062
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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.