filtration of $\sigma$-algebras

For an ordered set $T$, a filtration of $\sigma$-algebras (http://planetmath.org/SigmaAlgebra) ${({ℱ}_t)}_{t\in T}$ is a collection of $\sigma$-algebras on an underlying set $\mathrm{\Omega }$, satisfying ${ℱ}_s\subseteq {ℱ}_t$ for all in $T$. Here, $t$ is understood as the time variable, taking values in the index set $T$, and ${ℱ}_t$ represents the collection of all events observable up until time $t$. The index set is usually a subset of the real numbers, with common examples being $T={\mathbb{Z}}_{+}$ for discrete-time and $T={ℝ}_{+}$ for continuous-time scenarios.The collection ${({ℱ}_t)}_{t\in T}$ is a filtration on a measurable space $(\mathrm{\Omega },ℱ)$ if ${ℱ}_t\subseteq ℱ$ for every $t$. If, furthermore, there is a probability measure defined on the underlying measurable space then this gives a filtered probability space.The alternative notation $({ℱ}_t,t\in T)$ is often used for the filtration or, when the index set $T$ is clear from the context, simply $({ℱ}_t)$ or $𝐅$.

Filtrations are widely used for studying stochastic processes, where a process $X_t$ with time ranging over the set $T$ is said to be adapted to the filtration if $X_t$ is an ${ℱ}_t$-measurable random variable for each time $t$.

Conversely, any stochastic process ${(X_t)}_{t\in T}$ generates a filtration. Let ${ℱ}_t$ be the smallest $\sigma$-algebra with respect to which $X_s$ is measurable for all $s\le t$,

$${ℱ}_t=\sigma (X_s:s\le t).$$

This defines the smallest filtration to which $X$ is adapted, known as the natural filtration of $X$.

Given a filtration, there are various limiting $\sigma$-algebras which can be defined. The values at plus and minus infinity are

$${ℱ}_{\mathrm{\infty }}=\sigma \left(\bigcup _t{ℱ}_t\right),{ℱ}_{-\mathrm{\infty }}=\bigcap _t{ℱ}_t,$$

which satisfy ${ℱ}_{-\mathrm{\infty }}\subseteq {ℱ}_t\subseteq {ℱ}_{\mathrm{\infty }}$.In continuous-time, when the index set is an interval of the real numbers, the left and right limits can be defined at any time. They are,

except if $t$ is the maximum of $T$ it is often convenient to set ${ℱ}_{t+}={ℱ}_t$ or, if $t$ is the minimum, ${ℱ}_{t-}={ℱ}_t$. It is easily verified that ${ℱ}_s\subseteq {ℱ}_{s+}\subseteq {ℱ}_{t-}\subseteq {ℱ}_t$ for all times .Furthermore, $({ℱ}_{t+})$ and $({ℱ}_{t-})$ are themselves filtrations.

A filtration is said to be right-continuous if ${ℱ}_t={ℱ}_{t+}$ for every $t$ so, in particular, $({ℱ}_{t+})$ is always the smallest right-continuous filtration larger than $({ℱ}_t)$.