Finite filtration

Mathematical statement

Represent each time's sigma-algebra by a partition of the same finite sample space. F_s being contained in F_t means the later partition refines the earlier one. A filtration specifies information nesting only: a process is not automatically adapted, and no natural-filtration, independence or Markov claim follows.

Representative formula (exact TeX)

F(t)

Assumptions

• Every partition uses the same outcomes 0 through n−1; labels within a row indicate only block equivalence.
• Refinement is checked completely at every pair of adjacent times; finitely sampled events do not establish sigma-algebra inclusion.

Symbol scope

• F — finite filtration
• t — time index

Intuition, definition, and misconceptions

A finer partition represents more information.

Every later block must be contained in an earlier block.

An event is measurable exactly when it is a union of current blocks; adaptedness is checked separately.

An arbitrary partition sequence is not necessarily a filtration, and a filtration does not automatically make a process adapted, independent or Markov.

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

• Sigma-algebras and measure spaces — Each time's information is a sigma-algebra on the same space.
• Discrete-time stochastic process — A filtration supplies time-indexed information semantics.
• Sequence — The sigma-algebras are ordered in time.
• Measurable function — Adaptedness separately requires each time's variable to be measurable.

Verification boundary

Only complete finite partition sequences are handled; no natural-filtration or process-property claim 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

Together with conditional expectation, add martingales.

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.