Filtration (mathematics)

In mathematics, a filtration is an indexed set of subobjects of a given algebraic structure , with the index running over some index set that is a totally ordered set, subject to the condition that

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.[1]

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.

ExamplesEdit

AlgebraEdit

GroupsEdit

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 filtrationsEdit

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  .

When   is given the  -adic topology,   becomes a topological ring. If an  -module   is then given the  -adic topology, it becomes a topological  -module, relative to the topology given on  .

Rings and modules: ascending filtrationsEdit

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.

SetsEdit

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.

Measure theoryEdit

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

 

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)  , 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  ).[2][3][4]

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-algebrasEdit

Let   be a filtered probability space. A random variable   is a stopping time with respect to the filtration  , if   for all  . The stopping time  -algebra is now defined as

 .

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  .[5] 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  .[5]

It can be shown that   is  -measurable. However, simple examples[5] show that, in general,  . If   and   are stopping times on  , and   almost surely, then  

See alsoEdit

ReferencesEdit

  1. ^ Björk, Thomas (2005). "Appendix B". Arbitrage Theory in Continuous Time. ISBN 978-0-19-927126-9.
  2. ^ Péter Medvegyev (January 2009). "Stochastic Processes: A very simple introduction" (PDF). Retrieved June 25, 2012.
  3. ^ Claude Dellacherie (1979). Probabilities and Potential. Elsevier. ISBN 9780720407013.
  4. ^ George Lowther (November 8, 2009). "Filtrations and Adapted Processes". Retrieved June 25, 2012.
  5. ^ a b c Fischer, Tom (2013). "On simple representations of stopping times and stopping time sigma-algebras". Statistics and Probability Letters. 83 (1): 345–349. arXiv:1112.1603. doi:10.1016/j.spl.2012.09.024.