# Finite martingale

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

An integrable finite process adapted to the same filtration is a martingale when the next value's conditional expectation equals the current value almost surely at every step.

mko-martingale · definition · v0.10.0



## Mathematical statement

On one common finite probability space, completely check filtration nesting, adaptedness, integrability and E[X_(t+1)|F_t]=X_t for every t. On zero-mass information blocks conditional values are determined only almost surely. No independent-increment, Markov, optional-stopping or convergence conclusion follows.





## Representative formula (exact TeX)

M(X,F)





## Assumptions


- Exact rational masses, the partition filtration and process values belong to the same finite sample space.
- The main conditioning models use positive-mass blocks; zero mass records only the almost-sure boundary.





## Symbol scope


- M — finite martingale
- F — same filtration





## Intuition, definition, and misconceptions

Check filtration and adaptedness first.

For each time block compute the conditional mean of the next value.

Expectation invariance is a consequence, not the complete martingale definition.

Constant total expectation does not by itself define a martingale, and independent increments or optional stopping do not follow automatically.





## Adopted derivation guide

[A complete two-coin martingale audit](https://ocme.evemisslab.com/en/derivations/derivation-mko-martingale-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite filtration](https://ocme.evemisslab.com/en/mko/mko-finite-filtration/) — The information sequence must be completely nested.
- [Finite conditional expectation](https://ocme.evemisslab.com/en/mko/mko-conditional-expectation/) — Each step satisfies the martingale conditional-mean equation.
- [Discrete-time stochastic process](https://ocme.evemisslab.com/en/mko/mko-discrete-stochastic-process/) — X_t is time-indexed.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — Integrability and conditional means use expectation.
- [Measurable function](https://ocme.evemisslab.com/en/mko/mko-measurable-function/) — Adaptedness is checked at each time.





## Verification boundary

Only the specified finite-time model is handled; no convergence or stopping theorem is asserted.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add submartingales, stopping times and optional-stopping boundaries.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/martingale.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-martingale/index.md)