Filtration (mathematics)

In mathematics, a filtration formula_1 is an indexed family formula_2 of subobjects of a given algebraic structure formula_3, with the index formula_4 running over some totally ordered index set formula_5, subject to the condition that
If the index formula_4 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 formula_10 gaining in complexity with time. Hence, a process that is adapted to a filtration formula_1, 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 formula_10 be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication), that satisfy formula_13, 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 formula_10 be the whole formula_3, or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the formula_10 to formula_3 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 formula_18 in lieu of formula_19 (and, occasionally, formula_20 instead of formula_21). 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.
Examples.
Algebra.
Groups.
In algebra, filtrations are ordinarily indexed by formula_22, the set of natural numbers. A "filtration" of a group formula_23, is then a nested sequence formula_24 of normal subgroups of formula_23 (that is, for any formula_26 we have formula_27). Note that this use of the word "filtration" corresponds to our "descending filtration".
Given a group formula_23 and a filtration formula_24, there is a natural way to define a topology on formula_23, 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 formula_23 is defined to be open if it is a union of sets of the form formula_32, where formula_33 and formula_26 is a natural number.
The topology associated to a filtration on a group formula_23 makes formula_23 into a topological group.
The topology associated to a filtration formula_24 on a group formula_23 is Hausdorff if and only if formula_39.
If two filtrations formula_24 and formula_41 are defined on a group formula_23, then the identity map from formula_23 to formula_23, where the first copy of formula_23 is given the formula_24-topology and the second the formula_41-topology, is continuous if and only if for any formula_26 there is an formula_49 such that formula_50, 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 formula_51 and an formula_51-module formula_53, a "descending filtration" of formula_53 is a decreasing sequence of submodules formula_55. 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 formula_5-adic topology (or formula_57-adic, etc.). Let formula_51 be a commutative ring, and formula_5 an ideal of formula_51.
Given an formula_51-module formula_53, the sequence formula_63 of submodules of formula_53 forms a filtration of formula_53. The "formula_5-adic topology" on formula_53 is then the topology associated to this filtration. If formula_53 is just the ring formula_51 itself, we have defined the "formula_5-adic topology" on formula_51.
When formula_51 is given the formula_5-adic topology, formula_51 becomes a topological ring. If an formula_51-module formula_53 is then given the formula_5-adic topology, it becomes a topological formula_51-module, relative to the topology given on formula_51.
Rings and modules: ascending filtrations.
Given a ring formula_51 and an formula_51-module formula_53, an "ascending filtration" of formula_53 is an increasing sequence of submodules formula_55. In particular, if formula_51 is a field, then an ascending filtration of the formula_51-vector space formula_53 is an increasing sequence of vector subspaces of formula_53. 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 formula_89 corresponds to the ordering formula_90. 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.
Measure theory.
In measure theory, in particular in martingale theory and the theory of stochastic processes, a filtration is an increasing sequence of formula_91-algebras on a measurable space. That is, given a measurable space formula_92, a filtration is a sequence of formula_91-algebras formula_94 with formula_95 where each formula_96 is a non-negative real number and
The exact range of the "times" "formula_96" will usually depend on context: the set of values for formula_96 might be discrete or continuous, bounded or unbounded. For example,
Similarly, a filtered probability space (also known as a stochastic basis) formula_101, is a probability space equipped with the filtration formula_102 of its formula_91-algebra formula_1. A filtered probability space is said to satisfy the "usual conditions" if it is complete (i.e., formula_105 contains all formula_106-null sets) and right-continuous (i.e. formula_107 for all times formula_96).
It is also useful (in the case of an unbounded index set) to define formula_109 as the formula_91-algebra generated by the infinite union of the formula_111's, which is contained in formula_1:
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 formula_96". 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 formula_96, 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 formula_101 be a filtered probability space. A random variable formula_117 is a stopping time with respect to the filtration formula_118, if formula_119 for all formula_120. 
The "stopping time" formula_91-algebra is now defined as
It is not difficult to show that formula_123 is indeed a formula_91-algebra.
The set formula_123 encodes information up to the "random" time formula_126 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 formula_126 is formula_123. In particular, if the underlying probability space is finite (i.e. formula_1 is finite), the minimal sets of formula_123 (with respect to set inclusion) are given by the union over all formula_120 of the sets of minimal sets of formula_111 that lie in formula_133.
It can be shown that formula_126 is formula_123-measurable. However, simple examples show that, in general, formula_136. If formula_137 and formula_138 are stopping times on formula_101, and formula_140 almost surely, then formula_141