Filtration (probability theory)

Let ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ be a probability space and let ${\displaystyle I}$ be an index set with a total order ${\displaystyle \leq }$ (often ${\displaystyle \mathbb {N} }$, ${\displaystyle \mathbb {R} ^{+}}$, or a subset of ${\displaystyle \mathbb {R} ^{+}}$).

For every ${\displaystyle i\in I}$ let ${\displaystyle {\mathcal {F}}_{i}}$ be a sub-σ-algebra of ${\displaystyle {\mathcal {A}}}$. Then

${\displaystyle \mathbb {F}$ :=({\mathcal {F}}_{i})_{i\in I}}

is called a filtration, if ${\displaystyle {\mathcal {F}}_{k}\subseteq {\mathcal {F}}_{\ell }}$ for all ${\displaystyle k\leq \ell }$. So filtrations are families of σ-algebras that are ordered non-decreasingly.[1] If ${\displaystyle \mathbb {F} }$ is a filtration, then ${\displaystyle (\Omega ,{\mathcal {A}},\mathbb {F} ,P)}$ is called a filtered probability space.

Let ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ be a stochastic process on the probability space ${\displaystyle (\Omega ,{\mathcal {A}},P)}$. Let ${\displaystyle \sigma (X_{k}\mid k\leq n)}$ denote the σ-algebra generated by the random variables ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$. Then

${\displaystyle {\mathcal {F}}_{n}:=\sigma (X_{k}\mid k\leq n)}$

is a σ-algebra and ${\displaystyle \mathbb {F} =({\mathcal {F}}_{n})_{n\in \mathbb {N} }}$ is a filtration.

${\displaystyle \mathbb {F} }$ really is a filtration, since by definition all ${\displaystyle {\mathcal {F}}_{n}}$ are σ-algebras and

${\displaystyle \sigma (X_{k}\mid k\leq n)\subseteq \sigma (X_{k}\mid k\leq n+1).}$

This is known as the natural filtration of ${\displaystyle {\mathcal {A}}}$ with respect to ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$.

• Natural filtration
• Filtration (mathematics)
• Filter (mathematics)