> [!definition] > > Let $\cf$ be a $\sigma$-[[Sigma Algebra|field]]. A **filtration** over $\cf$ is an increasing family $\bracs{\cf_t: t \ge 0}$ of sub-$\sigma$-fields. > [!definition] > > Let $\bracs{\cf_t: t \ge 0}$ be a filtration and $t \ge 0$. Define > $ > \cf_{t^+} = \bigcap_{s > t}\cf_s > $ > as the intersection of all information from the strict immediate future, and > $ > \cf_{t^-} = \begin{cases} > \sigma\paren{\bigcup_{s < t}\cf_s} & t > 0 \\ > \cf_0 &t = 0 > \end{cases} > $ > as the combination of all information from the immediate past. The families $\bracs{\cf_{t^+}: t \ge 0}$ and $\bracs{\cf_{t^-}: t \ge 0}$ are both filtrations, with $\cf_{t^-} \subset \cf_{t} \subset \cf_{t^+}$. > > A filtration $\bracs{\cf_{t}: t \ge 0}$ is **left/right-continuous** if $\cf_t = \cf_{t^-}$/$\cf_t = \cf_{t^+}$ for all $t \ge 0$. > [!definition] > > Let $\bracs{X_t: t \ge 0}$ be a sequence of [[Random Variable|random variables]]. Then $\cf_t = \sigma(X_s)_{s \le t}$ is the filtration generated by/the natural filtration corresponding to $\bracs{X_t: t \ge 0}$. > > If $X$ has continuous sample paths, then its natural filtration is left-continuous. > [!definition] > > Let $\seqi{X}$ be a [[Stochastic Process|stochastic process]], and $\seqi{\cf}$ be a filtration. $\seqi{X}$ is $\seqi{\cf}$-**adapted** if each $X_i$ is $\cf_i$-[[Measurable Function|measurable]]. > [!definition] > > Let $(\Omega, \cf, \bracs{\cf_t: t \ge 0}, \bp)$ be a filtered probability space, and $\bracs{X_t: t \ge 0}$ be a continuous time [[Stochastic Process|stochastic process]]. $X$ is a **progressively measurable** with respect to $\bracs{\cf_t: t \ge 0}$ if the mapping > $ > [0, T] \times \Omega \to \real^d \quad (t, \omega) \mapsto X_t(\omega) > $ > is measurable with respect to the product $\sigma$-field $\cb([0, T]) \otimes \cf_t$. If $X$ is progressively measurable with respect to $\bracs{\cf_t: t \ge 0}$, then $X$ is also $\bracs{\cf_t: t \ge 0}$-adapted. > [!theorem] > > Let $(\Omega, \cf, \bracs{\cf_t: t \ge 0}, \bp)$ be a filtered probability space and $X$ be a continuous time stochastic process. $X$ has [[RCLL]] sample paths, then $X$ is $\bracs{\cf_t: t \ge 0}$-adapted if and only if $X$ is progressively measurable. > > *Proof*. Suppose that $X$ is $\bracs{\cf_t: t \ge 0}$-adapted. For any $t > 0$, $(s, \omega) \in [0, t] \times \Omega$, denote $\lceil s \rceil_n$ as the $n$-th order dyadic ceiling. By right-continuity, for every $s \in [0, t]$, $X_s(\omega) = \limv{n}X_{\lceil s \rceil_n}(\omega)$. Thus it's sufficient to show that each $X_{\lceil s \rceil_n}$ is measurable with respect to $\cb([0, t]) \otimes \cf_t$. Let $B \subset \real^d$ be a Borel set, then the preimages are cylinder sets, and we have the desired measurability.