adapted stochastic process

适应过程

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Other names / variants: adapted process · 适应列（离散时间）

Translation evidence: Literature usage

Meaning in context

Relative to a filtration $(\mathcal F_t)$, a process $(X_t)$ is adapted when $X_t$ is $\mathcal F_t$-measurable at every time $t$.

相对于给定信息流，每个时刻的变量都只使用该时刻已知的可测信息的过程。

Example

The partial sum $S_n$ of the first $n$ coin increments is adapted to their natural filtration.

Usage and distinctions

Adaptedness does not alone imply joint measurability or predictability; discrete predictable values are measurable one step earlier.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gregory F. Lawler — Notes on Probability§6.2 Martingales, measurability requirement in martingale 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：适应的、适应列；§6.5.1 连续时间鞅 · Checked 2026-10-14

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