Plain-English statement
A martingale has no predictable change: conditioned on current information, every future value has the current value as its expectation.
Mathematical statement
For s <= t, \\(\mathbb E[M_t\mid\mathcal F_s]=M_s\\) almost everywhere, with M adapted to F.
Intuition
The filtration represents available information; the conditional expectation identity says that information cannot reveal a systematic future drift.
Conditions
- a measured sample space and filtration
- a real process indexed by nonnegative time
- strong adaptedness and the conditional-expectation equality
Why these conditions cannot be dropped
- adaptedness prevents future-dependent process values
- almost-everywhere equality is the correct measure-theoretic equality for conditional expectation
- integrability is recovered by Mathlib from the martingale identity
Proof route
- reuse Mathlib.Martingale at NNReal time
- test the interface on constant processes under a finite measure
Lean interface notes
- the process codomain is Real
- Mathlib uses StronglyAdapted and AE conditional-expectation equality
- sample-path continuity is an additional property, not part of martingalehood
Read the mathematics first, then descend into Lean
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Proof architecture
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Lean statement
def IsChewiMartingale
{Ω : Type*} {m : MeasurableSpace Ω}
(process : ℝ≥0 → Ω → ℝ) (filtration : Filtration ℝ≥0 m)
(μ : Measure Ω) : Prop :=
MeasureTheory.Martingale process filtration μ
/-- A constant real process is a martingale under a finite measure. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Martingale.lean:21published source at 7bcd37294df1