Filtration

Remarks

Example: for a stochastic process $(X_t)_{t \in T}$, the natural filtration is $\mathcal{F}_t^X = \sigma(X_s : s \le t)$, the smallest $\sigma$-algebra for which all $X_s$ with $s \le t$ are measurable. If $s \le t$, every generator of $\mathcal{F}_s^X$ is a generator of $\mathcal{F}_t^X$, so clause 2 holds.

Non-example: let $X$ be a random variable that is not almost surely constant, so that $\sigma(X) \neq \set{\emptyset, \Omega}$. On $T = \set{0, 1}$, the family $\mathcal{F}_0 = \sigma(X)$, $\mathcal{F}_1 = \set{\emptyset, \Omega}$ fails clause 2, since $\mathcal{F}_0 \not\subseteq \mathcal{F}_1$: information would be forgotten.

In continuous time one often assumes the usual conditions: $\mathcal{F}_0$ contains all $P$-null sets of the completion of $\mathcal{F}$, and the filtration is right-continuous, that is $\mathcal{F}_t = \bigcap_{s > t} \mathcal{F}_s$ for every $t$.