Plain-English statement
A process is a local martingale when it becomes a centered martingale before every stop in an exhausting sequence.
Mathematical statement
M is adapted and there are increasing stopping times tau_n tending a.s. to infinity such that M_{t wedge tau_n}-M_0 is a martingale.
Intuition
Localization removes global integrability while retaining martingale structure on every bounded stochastic window.
Conditions
- adaptedness
- increasing stopping times
- a.s. divergence
- ordinary martingale behavior after stopping
Why these conditions cannot be dropped
- adaptedness prevents future information
- a.s. divergence ensures every finite sample-path time is eventually covered
Proof route
- this is the exact source definition
- localized Ito integrals are handled by Proposition 1.1.16
Lean interface notes
- uses Mathlib stoppedProcess
- the centered stopped process is passed to Mathlib Martingale
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Lean statement
def IsLocalMartingale
{Omega : Type*} {m : MeasurableSpace Omega}
(process : ℝ≥0 → Omega → ℝ) (filtration : Filtration ℝ≥0 m)
(mu : Measure Omega) : Prop :=
Adapted filtration process ∧
∃ tau : ℕ → Omega → WithTop ℝ≥0,
(∀ n, IsStoppingTime filtration (tau n)) ∧
Monotone tau ∧
(∀ᵐ omega ∂mu, Tendsto (fun n => tau n omega) atTop (𝓝 (⊤ : WithTop ℝ≥0))) ∧
∀ n, Martingale
(fun t omega => stoppedProcess process (tau n) t omega - process 0 omega)
filtration mu
end
end Localization
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheoryOpen AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Localization.lean:69published source at 7bcd37294df1