filtration in probability

σ代数流

Probability · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: filtration

Translation evidence: Literature usage

Meaning in context

An increasing family $(\mathcal F_t)$ of sub-sigma-algebras of a probability space's event sigma-algebra, indexed by ordered times: $s\le t$ implies $\mathcal F_s\subseteq\mathcal F_t$.

概率空间内按时间递增的子σ代数族，用来记录截至各时刻可获得的信息。

Example

For coin observations $\xi_j$, $\mathcal F_n=\sigma(\xi_1,\ldots,\xi_n)$ is their natural filtration.

Usage and distinctions

Completeness and right-continuity are additional conditions, not built into this definition; a dynamical flow is a different object.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gregory F. Lawler — Notes on Probability§6.2 Martingales, opening filtration definition · Checked 2026-10-14

Translation sources

1. 李东风 — 应用随机过程，第6章 鞅（北京大学）§6.1.1 定义，定义6.3 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.