Plain-English statement
An elementary adapted process is a finite step process whose coefficient on each interval is already known at that interval's left endpoint.
Mathematical statement
eta_t = sum_i H_i 1_{t in (t_i,t_(i+1)]} for a strict finite grid and bounded F_(t_i)-measurable H_i.
Intuition
The left-endpoint measurability prevents a coefficient from using future Brownian information.
Conditions
- strict finite time grid
- left-endpoint filtration measurability
- bounded coefficients
Why these conditions cannot be dropped
- strictness identifies non-overlapping steps
- adaptedness is needed for later martingale orthogonality
- boundedness gives the initial integrability class
Proof route
- store the source regularity in ElementaryAdaptedProcess
- define value by a Fin sum over half-open intervals
- unfold the definition
Lean interface notes
- Fin (n + 1) indexes endpoints
- Fin n indexes coefficients
- StronglyMeasurable is relative to filtration (times i.castSucc)
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Lean statement
theorem chewi_display_1_1_2
{Omega : Type*} {m : MeasurableSpace Omega}
{filtration : Filtration ℝ≥0 m} {n : ℕ}
(eta : ElementaryAdaptedProcess filtration n) (t : ℝ≥0) (omega : Omega) :
eta.value t omega =
∑ i, if eta.times i.castSucc < t ∧ t ≤ eta.times i.succ
then eta.coeff i omega else 0 :=
rfl
/-- The finite Brownian-increment sum used to define the Ito integral of an
elementary process at terminal time `T`. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:45published source at 7bcd37294df1