What is a Filtration?
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A probability space is a triple
\[(\Omega,\mathcal{F},\mathbb{P}),\]where:
- $\Omega$ is the set of possible outcomes.
- $\mathcal{F}$ is the collection of events to which probabilities can be assigned.
- $\mathbb{P}$ assigns a probability to each event in $\mathcal{F}$.
This model describes all the information that could eventually be available. In a process evolving over time, however, we usually do not know everything immediately. A filtration records how information is revealed.
What is a filtration?
A discrete-time filtration is a sequence of sigma-algebras
\[\{\mathcal{F}_n\}_{n\geq 0}\]such that
\[\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \cdots \subseteq \mathcal{F}.\]Each $\mathcal{F}_n$ represents the information available at time $n$.
The inclusion
\[\mathcal{F}_n\subseteq\mathcal{F}_{n+1}\]means that information is not forgotten. Every event that can be determined at time $n$ can still be determined at time $n+1$.
A two-toss example
Suppose that a fair coin is tossed twice. The sample space is
\[\Omega = \{ \mathrm{HH}, \mathrm{HT}, \mathrm{TH}, \mathrm{TT} \}.\]Before either toss is observed, the only events we can distinguish are the impossible event and the certain event. Thus,
\[\mathcal{F}_0 = \{ \varnothing,\Omega \}.\]After the first toss, we know whether the first result was heads or tails, but we do not yet know the second result. Therefore,
\[\mathcal{F}_1 = \left\{ \varnothing, \Omega, \{\mathrm{HH},\mathrm{HT}\}, \{\mathrm{TH},\mathrm{TT}\} \right\}.\]After both tosses, the exact outcome is known. We may therefore take
\[\mathcal{F}_2 = 2^\Omega,\]where $2^\Omega$ denotes the collection of all subsets of $\Omega$.
These sigma-algebras satisfy
\[\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \mathcal{F}_2.\]Hence, $\lbrace\mathcal{F}_0,\mathcal{F}_1,\mathcal{F}_2\rbrace$ is a filtration.
Adapted random processes
Let $X_n$ be the number of heads observed by time $n$. For example,
\[X_0=0,\]and
\[X_1 = \begin{cases} 1, & \text{if the first toss is heads},\\ 0, & \text{if the first toss is tails}. \end{cases}\]The value of $X_n$ can be determined using only the information in $\mathcal{F}_n$. In measure-theoretic language, $X_n$ is $\mathcal{F}_n$-measurable.
A process $\lbrace X_n \rbrace_{n \geq 0}$ is adapted to a filtration $\lbrace \mathcal{F}_n \rbrace_{n\geq 0}$ if
\[X_n \text{ is } \mathcal{F}_n\text{-measurable for every }n.\]Adaptedness formalizes the rule that a process cannot depend on information from the future. For instance, the indicator of the second toss being heads is not $\mathcal{F}_1$-measurable because the second toss has not yet been observed at time $1$.
Conditional expectation and current information
Let $X_2$ denote the total number of heads after both tosses. Before any toss is observed,
\[\mathbb{E}[X_2\mid\mathcal{F}_0]=1.\]After observing the first toss,
\[\mathbb{E}[X_2\mid\mathcal{F}_1] = \begin{cases} \dfrac{3}{2}, & \text{if the first toss is heads},\\[6pt] \dfrac{1}{2}, & \text{if the first toss is tails}. \end{cases}\]After both tosses,
\[\mathbb{E}[X_2\mid\mathcal{F}_2]=X_2.\]Thus, conditional expectation is a prediction based on the information currently available. As the filtration grows, the prediction incorporates more observations.
Continuous-time filtrations
In continuous time, a filtration is a family
\[\{\mathcal{F}_t\}_{t\geq 0}\]satisfying
\[\mathcal{F}_s\subseteq\mathcal{F}_t \qquad \text{whenever } 0\leq s\leq t.\]The interpretation is unchanged: $\mathcal{F}_t$ contains precisely the events that can be determined using the information available by time $t$.
Why filtrations matter
Filtrations make the phrase “information available up to the present” mathematically precise. They are fundamental in the study of:
- stochastic processes,
- martingales,
- stopping times,
- stochastic differential equations,
- stochastic control, and
- mathematical finance.
A filtration therefore does more than organize events. It provides the mathematical structure needed to distinguish the past and present from the unknown future.