- if in , then .
If the index 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 object gaining in complexity with time. Hence, a process that is adapted to a filtration , is also called non-anticipating, i.e. one that cannot see into the future.
Sometimes, as in a filtered algebra, there is instead the requirement that the be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication), that satisfy , 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 be the whole , or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the to 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 in lieu of (and, occasionally, instead of ). Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with cofiltrations (which consist of quotient objects rather than subobjects).
The concept dual to a filtration is called a cofiltration.
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.
In algebra, filtrations are ordinarily indexed by , the set of natural numbers. A filtration of a group , is then a nested sequence of normal subgroups of (that is, for any we have ). Note that this use of the word "filtration" corresponds to our "descending filtration".
Given a group and a filtration , there is a natural way to define a topology on , said to be associated to the filtration. A basis for this topology is the set of all translates 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 , where and is a natural number.
The topology associated to a filtration on a group makes into a topological group.
The topology associated to a filtration on a group is Hausdorff if and only if .
If two filtrations and are defined on a group , then the identity map from to , where the first copy of is given the -topology and the second the -topology, is continuous if and only if for any there is an such that , that is, if and only if the identity map is continuous at 1. In particular, the two filtrations define the same topology if and only if for any subgroup appearing in one there is a smaller or equal one appearing in the other.
Rings and modules: descending filtrations
Given a ring and an -module , a descending filtration of is a decreasing sequence of submodules . This is therefore a special case of the notion for groups, with the additional condition that the subgroups be submodules. The associated topology is defined as for groups.
An important special case is known as the -adic topology (or -adic, etc.). Let be a commutative ring, and an ideal of .
Given an -module , the sequence of submodules of forms a filtration of . The -adic topology on is then the topology associated to this filtration. If is just the ring itself, we have defined the -adic topology on .
Rings and modules: ascending filtrations
Given a ring and an -module , an ascending filtration of is an increasing sequence of submodules . In particular, if is a field, then an ascending filtration of the -vector space is an increasing sequence of vector subspaces of . Flags are one important class of such filtrations.
A maximal filtration of a set is equivalent to an ordering (a permutation) of the set. For instance, the filtration corresponds to the ordering . 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.
In measure theory, in particular in martingale theory and the theory of stochastic processes, a filtration is an increasing sequence of -algebras on a measurable space. That is, given a measurable space , a filtration is a sequence of -algebras with where each is a non-negative real number and
Similarly, a filtered probability space (also known as a stochastic basis) , is a probability space equipped with the filtration of its -algebra . A filtered probability space is said to satisfy the usual conditions if it is complete (i.e., contains all -null sets) and right-continuous (i.e. for all times ).
It is also useful (in the case of an unbounded index set) to define as the -algebra generated by the infinite union of the 's, which is contained in :
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.
Relation to stopping times: stopping time sigma-algebras
It is not difficult to show that is indeed a -algebra. The set 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 is . In particular, if the underlying probability space is finite (i.e. is finite), the minimal sets of (with respect to set inclusion) are given by the union over all of the sets of minimal sets of that lie in .
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