In mathematics, a filtration
Contents
- Groups
- Rings and modules descending filtrations
- Rings and modules ascending filtrations
- Sets
- Measure theory
- Relation to stopping times stopping time sigma algebras
- References
If the index 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 object Si gaining in complexity with time. Hence, a process that is adapted to a filtration
Sometimes, as in a filtered algebra, there is instead the requirement that the
Sometimes, filtrations are supposed to satisfy the additional requirement that the union of the
There is also the notion of a descending filtration, which is required to satisfy
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.
Groups
In algebra, filtrations are ordinarily indexed by N, the set of natural numbers. A filtration of a group G, is then a nested sequence Gn of normal subgroups of G (that is, for any n we have Gn+1 ⊆ Gn). Note that this use of the word "filtration" corresponds to our "descending filtration".
Given a group G and a filtration Gn, there is a natural way to define a topology on G, 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 G is defined to be open if it is a union of sets of the form aGn, where a∈G and n is a natural number.
The topology associated to a filtration on a group G makes G into a topological group.
The topology associated to a filtration Gn on a group G is Hausdorff if and only if ∩Gn = {1}.
If two filtrations Gn and G′n are defined on a group G, then the identity map from G to G, where the first copy of G is given the Gn-topology and the second the G′n-topology, is continuous if and only if for any n there is an m such that Gm ⊆G′n, 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 R and an R-module M, a descending filtration of M is a decreasing sequence of submodules Mn. 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 I-adic topology (or J-adic, etc.). Let R be a commutative ring, and I an ideal of R.
Given an R-module M, the sequence InM of submodules of M forms a filtration of M. The I-adic topology on M is then the topology associated to this filtration. If M is just the ring R itself, we have defined the I-adic topology on R.
When R is given the I-adic topology, R becomes a topological ring. If an R-module M is then given the I-adic topology, it becomes a topological R-module, relative to the topology given on R.
Rings and modules: ascending filtrations
Given a ring R and an R-module M, an ascending filtration of M is an increasing sequence of submodules Mn. In particular, if R is a field, then an ascending filtration of the R-vector space M is an increasing sequence of vector subspaces of M. Flags are one important class of such filtrations.
Sets
A maximal filtration of a set is equivalent to an ordering (a permutation) of the set. For instance, the filtration
Measure theory
In measure theory, in particular in martingale theory and the theory of stochastic processes, a filtration is an increasing sequence of σ-algebras on a measurable space. That is, given a measurable space
The exact range of the "times"
Similarly, a filtered probability space (also known as a stochastic basis)
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 t". 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 t, 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
Let
It is not difficult to show that
It can be shown that