Finite martingale

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Finite filtration — The information sequence must be completely nested.
• Finite conditional expectation — Each step satisfies the martingale conditional-mean equation.
• Discrete-time stochastic process — X_t is time-indexed.
• Expectation in finite models — Integrability and conditional means use expectation.
• 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.