Filtration (Probability)

In probability theory, a filtration is an increasing family of sigma-algebras used to describe how information accumulates over time. It provides the information structure relative to which a stochastic process is observed, predictions are made, and decisions can be taken without using information not yet available. Filtrations are fundamental to martingales, stopping times, and stochastic integration. (psdey.web.illinois.edu)

Definition and interpretation

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $T$ be an ordered time set, commonly $\{0,1,2,\ldots\}$ or $[0,\infty)$ . A filtration is a family

\mathbb F=(\mathcal F_t)_{t\in T}

of sub-sigma-algebras of $\mathcal F$ such that

\mathcal F_s\subseteq\mathcal F_t \qquad\text{whenever }s\le t.

The tuple $(\Omega,\mathcal F,\mathbb F,\mathbb P)$ is called a filtered probability space. (psdey.web.illinois.edu)

An event belongs to $\mathcal F_t$ when its occurrence can be determined from the information available at time $t$ . A real-valued random variable is $\mathcal F_t$ -measurable when its value is determined by that information. The inclusion condition expresses retention of past information: information available earlier remains available later. The family need not increase strictly, and it may already contain substantial information at time zero. (maths.dur.ac.uk)

The filtration is an information structure, not itself a probability distribution or a realized sequence of observations. Its definition requires only a measurable space; a probability measure becomes necessary for conditional expectations, independence, and other probabilistic properties. (stats.libretexts.org)

Natural filtrations and examples

The natural filtration of a process $X=(X_t)_{t\in T}$ is

\mathcal F_t^X=\sigma(X_s:s\le t),

where $\sigma(\cdot)$ denotes the sigma-algebra generated by the indicated random variables. It represents precisely the information supplied by observing the process up to and including time $t$ . (psdey.web.illinois.edu)

For example, let $C_1,C_2,\ldots$ be independent fair coin tosses, with values $0$ and $1$ . Set

\mathcal F_0=\{\varnothing,\Omega\}, \qquad \mathcal F_n=\sigma(C_1,\ldots,C_n).

At time $n$ , the first $n$ results are known, but subsequent results are not. The event “the first toss is heads” belongs to $\mathcal F_1$ and every later sigma-algebra. The event “the second toss is heads” belongs to $\mathcal F_2$ , but not to $\mathcal F_1$ . In a finite model, this can be pictured as successive refinement of a partition of possible outcomes. (maths.dur.ac.uk)

At the opposite extremes, the constant filtration $\mathcal F_t=\{\varnothing,\Omega\}$ supplies no nontrivial information, while $\mathcal F_t=\mathcal F$ makes every measurable event available from the beginning. Consequently, the formal definition alone does not guarantee that a filtration represents only past observations of a particular process. (stats.libretexts.org)

Adaptedness and stronger measurability conditions

A process $X$ is adapted to $\mathbb F$ if $X_t$ is $\mathcal F_t$ -measurable for every $t$ . Equivalently,

\mathcal F_t^X\subseteq\mathcal F_t \qquad\text{for every }t.

Thus, the natural filtration is the smallest filtration to which $X$ is adapted. Adaptedness means that the process is observable using the designated information; it does not mean that its future values are predictable. (stats.libretexts.org)

In continuous time, adaptedness alone does not ensure the joint measurability needed for many constructions. A real-valued process is progressively measurable if, for every $t$ , the restriction

(s,\omega)\longmapsto X_s(\omega), \qquad 0\le s\le t,

is measurable with respect to

\mathcal B([0,t])\otimes\mathcal F_t,

where $\mathcal B([0,t])$ is the Borel sigma-algebra. Progressive measurability implies adaptedness. Adapted processes with right-continuous or left-continuous paths are progressively measurable. (maths.ox.ac.uk)

A predictable process satisfies a stronger information condition. In discrete time, $H_n$ is predictable when it is $\mathcal F_{n-1}$ -measurable for $n\ge1$ . In continuous time, predictability is defined through the sigma-algebra generated by left-continuous adapted processes, with the usual convention for time zero. Every continuous adapted process is predictable. Predictability is central to stochastic integration, where an integrand must not exploit an increment before it occurs. (web.math.wisc.edu)

Right-continuity, completeness, and the usual conditions

For a continuous-time filtration, define

\mathcal F_{t+}=\bigcap_{u>t}\mathcal F_u.

The filtration is right-continuous if

\mathcal F_t=\mathcal F_{t+}

for every $t$ . This says that information available at every time strictly after $t$ , however close to $t$ , is already included at $t$ . It is a property of the sigma-algebras, not a statement that sample paths are continuous. (psdey.web.illinois.edu)

A filtration is complete when $\mathcal F_0$ contains every subset of every $\mathbb P$ -null event in the ambient probability space. If necessary, the ambient space is completed as well. Completeness and right-continuity together are called the usual conditions, or usual hypotheses. They are additional assumptions, not part of the basic definition. A usual augmentation incorporates null subsets and right-continuity into a given filtration. (maths.ox.ac.uk)

These conditions support regularity results. In particular, a real-valued martingale on a filtered probability space satisfying the usual conditions admits a modification whose paths are right-continuous and have left limits, commonly called càdlàg paths. (maths.ox.ac.uk)

Conditional expectation and martingales

For an integrable random variable $Y$ ,

\mathbb E[Y\mid\mathcal F_t]

is its conditional expectation given the information available at time $t$ . An adapted process $M$ , with $\mathbb E|M_t|<\infty$ , is a martingale relative to $\mathbb F$ if

\mathbb E[M_t\mid\mathcal F_s]=M_s, \qquad s\le t,

with equality almost surely. The filtration is therefore part of the martingale property: it specifies what information is used to evaluate the expected future value. (math.uchicago.edu)

A basic construction is

M_t=\mathbb E[Y\mid\mathcal F_t].

The tower property of conditional expectation makes $M$ a martingale. In discrete time, if

\mathcal F_\infty=\sigma\!\left(\bigcup_{n\ge0}\mathcal F_n\right),

then

\mathbb E[Y\mid\mathcal F_n] \longrightarrow \mathbb E[Y\mid\mathcal F_\infty]

almost surely and in $L^1$ . This describes the limiting estimate obtained from all information eventually revealed. (tamuz.caltech.edu)

Stopping times

A random time $\tau:\Omega\to[0,\infty]$ is a stopping time relative to $\mathbb F$ if

\{\tau\le t\}\in\mathcal F_t \qquad\text{for every }t\ge0.

By time $t$ , one can determine whether stopping has already occurred. One need not know the stopping time in advance. (psdey.web.illinois.edu)

The first head in the coin-toss example is a stopping time: whether it has occurred by toss $n$ depends only on the first $n$ results. First entry times of discrete-time adapted processes into measurable sets are stopping times. Continuous-time versions require appropriate measurability or path regularity assumptions; for example, the first hitting time of a closed set by a continuous adapted process is a stopping time. (stats.libretexts.org)

Information available at a stopping time is represented by

\mathcal F_\tau = \{A\in\mathcal F: A\cap\{\tau\le t\}\in\mathcal F_t \text{ for every }t\ge0\}.

This sigma-algebra allows conditional expectations and martingale results to be formulated at random rather than deterministic times. (psdey.web.illinois.edu)

Dependence on the chosen information structure

Enlarging a filtration preserves adaptedness but need not preserve the martingale property. For Brownian motion $B$ to be Brownian motion relative to $\mathbb F$ , it must be adapted and each future increment $B_t-B_s$ must be independent of $\mathcal F_s$ for $s<t$ . Adaptedness by itself is insufficient. (math.uchicago.edu)

For illustration, enlarge Brownian motion’s natural filtration by revealing $B_T$ at time zero, for a fixed $T>0$ . The process remains adapted, but it cannot remain a martingale: $B_T$ is now measurable at time zero, so

\mathbb E[B_T\mid\mathcal G_0]=B_T,

whereas the martingale identity would require this conditional expectation to equal $B_0=0$ . This is a direct consequence of the martingale definition and illustrates why “fairness” depends on information. (math.uchicago.edu)

In stochastic integration and mathematical models of financial markets, a filtration specifies the observations that may determine an integrand or trading strategy. Requiring decisions to be measurable with respect to current or preceding information formalizes the exclusion of foresight. Changing the filtration can therefore change which strategies are admissible, even when the underlying random variables remain unchanged. (web.math.wisc.edu)

References

Lecture 6: Filtrations and Stopping Timespsdey.web.illinois.edu
11: Filtrations and Stopping Timesstats.libretexts.org
Chapter 4: Filtrations, Conditional Expectation, and Martingalesmaths.dur.ac.uk
Continuous Martingales and Stochastic Calculusmaths.ox.ac.uk
Basics of Stochastic Analysisweb.math.wisc.edu
Brownian Motion and Stochastic Calculusmath.uchicago.edu
Lecture Notes on Probabilitytamuz.caltech.edu