IsLocalizingSequence · Samplinglib

Plain-English statement

A localizing sequence stops an integrand where it is square-integrable and exhausts the requested time interval almost surely.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

tau_n are increasing stopping times, E integral_0^T |eta_t 1_{t<=tau_n}|^2 dt is finite, and tau_n tends to T almost surely.

Intuition

Each stopped problem lies in the global L2 theory; the almost-sure limit recovers the original finite interval.

Conditions

progressive measurability
stopping-time measurability
monotonicity
finite stopped L2 norms
a.s. convergence

Why these conditions cannot be dropped

progressiveness makes the stochastic integrand admissible
the L2 condition permits the existing Ito integral
the limit ensures the stops exhaust the target interval

Proof route

this is the exact source definition
the canonical sequence is proved separately

Lean interface notes

the L2 quantity is an ENNReal iterated lintegral
NNReal time uses the pullback of real Lebesgue measure

Lean learning studio · mathematics → formal proof

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Lean statement

def IsLocalizingSequence
    {Omega : Type*} {m : MeasurableSpace Omega}
    (eta : ℝ≥0 → Omega → ℝ) (filtration : Filtration ℝ≥0 m)
    (mu : Measure Omega) (T : ℝ≥0)
    (tau : ℕ → Omega → WithTop ℝ≥0) : Prop :=
  IsStronglyProgressive filtration eta ∧
    (∀ n, IsStoppingTime filtration (tau n)) ∧
    Monotone tau ∧
    (∀ n,
      (∫⁻ omega, ∫⁻ t in Icc (0 : ℝ≥0) T,
        ENNReal.ofReal ((stoppedIntegrand eta (tau n) t omega) ^ 2) ∂nnrealLebesgue ∂mu) < ∞) ∧
    ∀ᵐ omega ∂mu, Tendsto (fun n => tau n omega) atTop (𝓝 (T : WithTop ℝ≥0))

/-- Chewi Definition 1.1.15: an adapted process is a local martingale when a
monotone sequence of stopping times tends to infinity almost surely and every
stopped, initially centered process is a martingale.

The limit is the *topological* neighborhood of `⊤` in `WithTop ℝ≥0`.  Using
the order filter `atTop` as the codomain would be too strong here because
`WithTop ℝ≥0` has a greatest element: that filter would force the stopping
times to be eventually equal to `⊤`, rather than allowing finite stopping
times to diverge to infinity as in Chewi's definition. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Localization.lean:47published source at 77184245109a