# 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 $\mathrm{\Omega}$, satisfying ${\mathcal{F}}_{s}\subseteq {\mathcal{F}}_{t}$ for all $$ 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^{} $(\mathrm{\Omega},\mathcal{F})$ 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 $\mathbf{F}$.

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 $s\le t$,

$${\mathcal{F}}_{t}=\sigma ({X}_{s}:s\le t).$$ |

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

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

which satisfy ${\mathcal{F}}_{-\mathrm{\infty}}\subseteq {\mathcal{F}}_{t}\subseteq {\mathcal{F}}_{\mathrm{\infty}}$. 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 ${\mathcal{F}}_{t+}={\mathcal{F}}_{t}$ 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 $$. 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 |
---|---|

Canonical name | FiltrationOfsigmaalgebras |

Date of creation | 2013-03-22 18:37:13 |

Last modified on | 2013-03-22 18:37:13 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 5 |

Author | gel (22282) |

Entry type | Definition |

Classification | msc 60G05 |

Synonym | filtration of sigma-algebras |

Related topic | FilteredProbabilitySpace |

Related topic | Filtration |

Defines | natural filtration |