Filtration (mathematics)

See: Filtered algebra

In algebra, filtrations are ordinarily indexed by ${\displaystyle \mathbb {N} }$, the set of natural numbers. A filtration of a group ${\displaystyle G}$, is then a nested sequence ${\displaystyle G_{n}}$ of normal subgroups of ${\displaystyle G}$ (that is, for any ${\displaystyle n}$ we have ${\displaystyle G_{n+1}\subseteq G_{n}}$). Note that this use of the word "filtration" corresponds to our "descending filtration".

Given a group ${\displaystyle G}$ and a filtration ${\displaystyle G_{n}}$, there is a natural way to define a topology on ${\displaystyle G}$, 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 ${\displaystyle G}$ is defined to be open if it is a union of sets of the form ${\displaystyle aG_{n}}$, where ${\displaystyle a\in G}$ and ${\displaystyle n}$ is a natural number.

The topology associated to a filtration on a group ${\displaystyle G}$ makes ${\displaystyle G}$ into a topological group.

The topology associated to a filtration ${\displaystyle G_{n}}$ on a group ${\displaystyle G}$ is Hausdorff if and only if ${\displaystyle \bigcap G_{n}=\{1\}}$.

If two filtrations ${\displaystyle G_{n}}$ and ${\displaystyle G'_{n}}$ are defined on a group ${\displaystyle G}$, then the identity map from ${\displaystyle G}$ to ${\displaystyle G}$, where the first copy of ${\displaystyle G}$ is given the ${\displaystyle G_{n}}$-topology and the second the ${\displaystyle G'_{n}}$-topology, is continuous if and only if for any ${\displaystyle n}$ there is an ${\displaystyle m}$ such that ${\displaystyle G_{m}\subseteq G'_{n}}$, 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.

Given a ring ${\displaystyle R}$ and an ${\displaystyle R}$-module ${\displaystyle M}$, a descending filtration of ${\displaystyle M}$ is a decreasing sequence of submodules ${\displaystyle M_{n}}$. 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 ${\displaystyle I}$-adic topology (or ${\displaystyle J}$-adic, etc.): Let ${\displaystyle R}$ be a commutative ring, and ${\displaystyle I}$ an ideal of ${\displaystyle R}$. Given an ${\displaystyle R}$-module ${\displaystyle M}$, the sequence ${\displaystyle I^{n}M}$ of submodules of ${\displaystyle M}$ forms a filtration of ${\displaystyle M}$. The ${\displaystyle I}$-adic topology on ${\displaystyle M}$ is then the topology associated to this filtration. If ${\displaystyle M}$ is just the ring ${\displaystyle R}$ itself, we have defined the ${\displaystyle I}$-adic topology on ${\displaystyle R}$.

When ${\displaystyle R}$ is given the ${\displaystyle I}$-adic topology, ${\displaystyle R}$ becomes a topological ring. If an ${\displaystyle R}$-module ${\displaystyle M}$ is then given the ${\displaystyle I}$-adic topology, it becomes a topological ${\displaystyle R}$-module, relative to the topology given on ${\displaystyle R}$.

Given a ring ${\displaystyle R}$ and an ${\displaystyle R}$-module ${\displaystyle M}$, an ascending filtration of ${\displaystyle M}$ is an increasing sequence of submodules ${\displaystyle M_{n}}$. In particular, if ${\displaystyle R}$ is a field, then an ascending filtration of the ${\displaystyle R}$-vector space ${\displaystyle M}$ is an increasing sequence of vector subspaces of ${\displaystyle M}$. 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 ${\displaystyle \{0\}\subseteq \{0,1\}\subseteq \{0,1,2\}}$ corresponds to the ordering ${\displaystyle (0,1,2)}$. 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 ${\displaystyle \sigma }$-algebras on a measurable space. That is, given a measurable space ${\displaystyle (\Omega ,{\mathcal {F}})}$, a filtration is a sequence of ${\displaystyle \sigma }$-algebras ${\displaystyle \{{\mathcal {F}}_{t}\}_{t\geq 0}}$ with ${\displaystyle {\mathcal {F}}_{t}\subseteq {\mathcal {F}}}$ where each ${\displaystyle t}$ is a non-negative real number and

${\displaystyle t_{1}\leq t_{2}\implies {\mathcal {F}}_{t_{1}}\subseteq {\mathcal {F}}_{t_{2}}.}$

The exact range of the "times" ${\displaystyle t}$ will usually depend on context: the set of values for ${\displaystyle t}$ might be discrete or continuous, bounded or unbounded. For example,

${\displaystyle t\in \{0,1,\dots ,N\},\mathbb {N} _{0},[0,T]{\mbox{ or }}[0,+\infty ).}$

Similarly, a filtered probability space (also known as a stochastic basis) ${\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)}$, is a probability space equipped with the filtration ${\displaystyle \left\{{\mathcal {F}}_{t}\right\}_{t\geq 0}}$ of its ${\displaystyle \sigma }$-algebra ${\displaystyle {\mathcal {F}}}$. A filtered probability space is said to satisfy the usual conditions if it is complete (i.e., ${\displaystyle {\mathcal {F}}_{0}}$ contains all ${\displaystyle \mathbb {P} }$-null sets) and right-continuous (i.e. ${\displaystyle {\mathcal {F}}_{t}={\mathcal {F}}_{t+}:=\bigcap _{s>t}{\mathcal {F}}_{s}}$ for all times ${\displaystyle t}$).[2][3][4]

It is also useful (in the case of an unbounded index set) to define ${\displaystyle {\mathcal {F}}_{\infty }}$ as the ${\displaystyle \sigma }$-algebra generated by the infinite union of the ${\displaystyle {\mathcal {F}}_{t}}$'s, which is contained in ${\displaystyle {\mathcal {F}}}$:

${\displaystyle {\mathcal {F}}_{\infty }=\sigma \left(\bigcup _{t\geq 0}{\mathcal {F}}_{t}\right)\subseteq {\mathcal {F}}.}$

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 ${\displaystyle t}$". 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 ${\displaystyle t}$, 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.

Let ${\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)}$ be a filtered probability space. A random variable ${\displaystyle \tau$ :\Omega \rightarrow [0,\infty ]} is a stopping time with respect to the filtration ${\displaystyle \left\{{\mathcal {F}}_{t}\right\}_{t\geq 0}}$, if ${\displaystyle \{\tau \leq t\}\in {\mathcal {F}}_{t}}$ for all ${\displaystyle t\geq 0}$. The stopping time ${\displaystyle \sigma }$-algebra is now defined as

${\displaystyle {\mathcal {F}}_{\tau }:=\{A\in {\mathcal {F}}\vert \forall t\geq 0\colon A\cap \{\tau \leq t\}\in {\mathcal {F}}_{t}\}}$.

It is not difficult to show that ${\displaystyle {\mathcal {F}}_{\tau }}$ is indeed a ${\displaystyle \sigma }$-algebra. The set ${\displaystyle {\mathcal {F}}_{\tau }}$ encodes information up to the random time ${\displaystyle \tau }$ 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 ${\displaystyle \tau }$ is ${\displaystyle {\mathcal {F}}_{\tau }}$.[5] In particular, if the underlying probability space is finite (i.e. ${\displaystyle {\mathcal {F}}}$ is finite), the minimal sets of ${\displaystyle {\mathcal {F}}_{\tau }}$ (with respect to set inclusion) are given by the union over all ${\displaystyle t\geq 0}$ of the sets of minimal sets of ${\displaystyle {\mathcal {F}}_{t}}$ that lie in ${\displaystyle \{\tau =t\}}$.[5]

It can be shown that ${\displaystyle \tau }$ is ${\displaystyle {\mathcal {F}}_{\tau }}$-measurable. However, simple examples[5] show that, in general, ${\displaystyle \sigma (\tau )\neq {\mathcal {F}}_{\tau }}$. If ${\displaystyle \tau _{1}}$ and ${\displaystyle \tau _{2}}$ are stopping times on ${\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)}$, and ${\displaystyle \tau _{1}\leq \tau _{2}}$ almost surely, then ${\displaystyle {\mathcal {F}}_{\tau _{1}}\subseteq {\mathcal {F}}_{\tau _{2}}.}$