Project documentation
Apply the optional stopping theorem to get equation 4. Note that T1 Space is needed to make sure that mesh ι n has order topology.
Exact Lean statement
lemma equation4 {ι Ω E : Type*} [TopologicalSpace ι] [T1Space ι] [SecondCountableTopology ι]
[LinearOrder ι] [OrderBot ι] [OrderTop ι] {mΩ : MeasurableSpace Ω}
{P : Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {S : ι → Ω → E}
{𝓕 : Filtration ι mΩ} {n : ℕ} [SigmaFiniteFiltration P 𝓕] (hstop : S ⊤ =ᶠ[ae P] 0)
{τ : Ω → WithTop (mesh ι n)} (hτ : ∀ ω, τ ω ≤ WithTop.some (⊤ : mesh ι n))
(hτs : IsStoppingTime (meshFiltration 𝓕 n) τ) :
stoppedValue (S ∘ Subtype.val) τ =ᵐ[P]
-P[(predictableSeqTop S 𝓕 P n) | hτs.measurableSpace] +
stoppedValue (predictablePart (S ∘ Subtype.val) (meshFiltration 𝓕 n) P) τFormal artifact
Lean source
lemma equation4 {ι Ω E : Type*} [TopologicalSpace ι] [T1Space ι] [SecondCountableTopology ι] [LinearOrder ι] [OrderBot ι] [OrderTop ι] {mΩ : MeasurableSpace Ω} {P : Measure Ω} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {S : ι → Ω → E} {𝓕 : Filtration ι mΩ} {n : ℕ} [SigmaFiniteFiltration P 𝓕] (hstop : S ⊤ =ᶠ[ae P] 0) {τ : Ω → WithTop (mesh ι n)} (hτ : ∀ ω, τ ω ≤ WithTop.some (⊤ : mesh ι n)) (hτs : IsStoppingTime (meshFiltration 𝓕 n) τ) : stoppedValue (S ∘ Subtype.val) τ =ᵐ[P] -P[(predictableSeqTop S 𝓕 P n) | hτs.measurableSpace] + stoppedValue (predictablePart (S ∘ Subtype.val) (meshFiltration 𝓕 n) P) τ := by grw [← condExp_neg, ← martingaleSeqTop_eq_neg_predictableSeqTop 𝓕 hstop] simp only [martingaleSeqTop] grw [← (martingale_martingalePart (S ∘ Subtype.val) (meshFiltration 𝓕 n) P).stoppedValue_ae_eq_condExp_of_le_const hτs hτ, ← stoppedValue.add, _root_.martingalePart_add_predictablePart]- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/DoobMeyer.lean:286-299
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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.