running over some totally ordered index set
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
, 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
, or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the
Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated.
There is also the notion of a descending filtration, which is required to satisfy
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.
A basis for this topology is the set of all cosets of subgroups appearing in the filtration, that is, a subset of
is defined to be open if it is a union of sets of the form
This is therefore a special case of the notion for groups, with the additional condition that the subgroups be submodules.
is an increasing sequence of vector subspaces of
From the point of view of the field with one element, an ordering on a set corresponds to a maximal flag (a filtration on a vector space), considering a set to be a vector space over the field with one element.
is a non-negative real number and The exact range of the "times"
will usually depend on context: the set of values for
might be discrete or continuous, bounded or unbounded.
For example, Similarly, a filtered probability space (also known as a stochastic basis)
A filtered probability space is said to satisfy the usual conditions if it is complete (i.e.,
[2][3][4] It is also useful (in the case of an unbounded index set) to define
: A σ-algebra defines the set of events that can be measured, which in a probability context is equivalent to events that can be discriminated, or "questions that can be answered at time
Therefore, a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of information.
A typical example is in mathematical finance, where a filtration represents the information available up to and including each time
, and is more and more precise (the set of measurable events is staying the same or increasing) as more information from the evolution of the stock price becomes available.
encodes information up to the random time
in the sense that, if the filtered probability space is interpreted as a random experiment, the maximum information that can be found out about it from arbitrarily often repeating the experiment until the random time
[5] In particular, if the underlying probability space is finite (i.e.
However, simple examples[5] show that, in general,