Filtration (mathematics) explained
In mathematics, a filtration
is an
indexed family
of
subobjects of a given
algebraic structure
, with the index
running over some
totally ordered index 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 structure
gaining in complexity with time. Hence, a process that is
adapted to a filtration
is also called
non-anticipating, because it 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 only
, 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 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
Sets
Farey Sequence
Algebra
Algebras
See: Filtered algebra
Groups
See also: Length function.
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
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
, 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 filtration). 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 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.
Sets
A maximal filtration of a set is equivalent to an ordering (a permutation) of the set. For instance, the filtration
\{0\}\subseteq\{0,1\}\subseteq\{0,1,2\}
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 theory
See main article: article and Filtration (probability theory). 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
t1\leqt2\implies
\subseteq
.
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,t\in\{0,1,...,N\},N0,[0,T]or[0,+infty).
Similarly, a filtered probability space (also known as a stochastic basis)
\left(\Omega,l{F},\left\{l{F}t\right\}t\geq,P\right)
, is a
probability space equipped with the filtration
\left\{l{F}t\right\}t\geq
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
:
l{F}infty=\sigma\left(cuptl{F}t\right)\subseteql{F}.
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
See main article: article and σ-Algebra of τ-past. Let
\left(\Omega,l{F},\left\{l{F}t\right\}t\geq,P\right)
be a filtered probability space. A random variable
is a
stopping time with respect to the filtration
\left\{l{F}t\right\}t\geq
, if
for all
. The
stopping time
-algebra is now defined as
l{F}\tau:=\{A\inl{F}\vert\forallt\geq0\colonA\cap\{\tau\leqt\}\inl{F}t\}
.
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
\left(\Omega,l{F},\left\{l{F}t\right\}t\geq,P\right)
, and
almost surely, then
See also
References
- Book: Øksendal, Bernt K. . Bernt Øksendal . Stochastic Differential Equations: An Introduction with Applications . Springer. Berlin . 2003 . 978-3-540-04758-2.
Notes and References
- Book: Björk, Thomas. 2005. Arbitrage Theory in Continuous Time. 978-0-19-927126-9. Appendix B.
- Web site: Stochastic Processes: A very simple introduction. Péter Medvegyev. January 2009. June 25, 2012. April 3, 2015. https://web.archive.org/web/20150403125546/http://medvegyev.uni-corvinus.hu/St1.pdf. dead.
- Book: Probabilities and Potential. Claude Dellacherie. Elsevier. 1979. 9780720407013.
- Web site: Filtrations and Adapted Processes. George Lowther. November 8, 2009. June 25, 2012.
- Fischer. Tom. On simple representations of stopping times and stopping time sigma-algebras. Statistics and Probability Letters. 2013. 83. 1. 345–349. 10.1016/j.spl.2012.09.024. 1112.1603.