# filtration of $\sigma$-algebras

For an ordered set $T$, a filtration of $\sigma$-algebras (http://planetmath.org/SigmaAlgebra) $(\mathcal{F}_{t})_{t\in T}$ is a collection of $\sigma$-algebras on an underlying set $\Omega$, satisfying $\mathcal{F}_{s}\subseteq\mathcal{F}_{t}$ for all $s in $T$. Here, $t$ is understood as the time variable, taking values in the index set $T$, and $\mathcal{F}_{t}$ 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 $T=\mathbb{R}_{+}$ for continuous-time scenarios. The collection $(\mathcal{F}_{t})_{t\in T}$ is a filtration on a measurable space $(\Omega,\mathcal{F})$ if $\mathcal{F}_{t}\subseteq\mathcal{F}$ 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 $(\mathcal{F}_{t},t\in T)$ is often used for the filtration or, when the index set $T$ is clear from the context, simply $(\mathcal{F}_{t})$ or ${\bf F}$.

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 $\mathcal{F}_{t}$-measurable random variable for each time $t$.

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

 $\mathcal{F}_{t}=\sigma\left(X_{s}:s\leq t\right).$

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

 $\mathcal{F}_{\infty}=\sigma\left(\bigcup_{t}\mathcal{F}_{t}\right),\ \mathcal{% F}_{-\infty}=\bigcap_{t}\mathcal{F}_{t},$

which satisfy $\mathcal{F}_{-\infty}\subseteq\mathcal{F}_{t}\subseteq\mathcal{F}_{\infty}$. 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,

 $\mathcal{F}_{t+}=\bigcap_{s>t}\mathcal{F}_{s},\ \mathcal{F}_{t-}=\sigma\left(% \bigcup_{s

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

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

Title filtration of $\sigma$-algebras FiltrationOfsigmaalgebras 2013-03-22 18:37:13 2013-03-22 18:37:13 gel (22282) gel (22282) 5 gel (22282) Definition msc 60G05 filtration of sigma-algebras FilteredProbabilitySpace Filtration natural filtration