Filtration (mathematics)

In mathematics, a filtration ${\displaystyle {\mathcal {F}}}$ is an indexed family ${\displaystyle (S_{i})_{i\in I}}$ of subobjects of a given algebraic structure ${\displaystyle S}$, with the index ${\displaystyle i}$ running over some totally ordered index set ${\displaystyle I}$, subject to the condition that

if ${\displaystyle i\leq j}$ in ${\displaystyle I}$, then ${\displaystyle S_{i}\subseteq S_{j}}$.

If the index ${\displaystyle i}$ is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure ${\displaystyle S_{i}}$ gaining in complexity with time. Hence, a process that is adapted to a filtration ${\displaystyle {\mathcal {F}}}$ is also called non-anticipating, because it cannot "see into the future".^[1]

Sometimes, as in a filtered algebra, there is instead the requirement that the ${\displaystyle S_{i}}$ be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication) that satisfy only ${\displaystyle S_{i}\cdot S_{j}\subseteq S_{i+j}}$, where the index set is the natural numbers; this is by analogy with a graded algebra.

Sometimes, filtrations are supposed to satisfy the additional requirement that the union of the ${\displaystyle S_{i}}$ be the whole ${\displaystyle S}$, or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the ${\displaystyle S_{i}}$ to ${\displaystyle S}$ is an isomorphism. Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated. This article does not impose this requirement.

There is also the notion of a descending filtration, which is required to satisfy ${\displaystyle S_{i}\supseteq S_{j}}$ in lieu of ${\displaystyle S_{i}\subseteq S_{j}}$ (and, occasionally, ${\displaystyle \bigcap _{i\in I}S_{i}=0}$ instead of ${\displaystyle \bigcup _{i\in I}S_{i}=S}$). Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with the dual notion of cofiltrations (which consist of quotient objects rather than subobjects).

Filtrations are widely used in abstract algebra, homological algebra (where they are related in an important way to spectral sequences), and in measure theory and probability theory for nested sequences of σ-algebras. In functional analysis and numerical analysis, other terminology is usually used, such as scale of spaces or nested spaces.