Filtration (probability theory)
A filtration is an increasing family of σ-algebras representing the accumulation of information over time in a probability space. Given a probability space ((\Omega,\mathcal F,\mathbb P)) and an ordered index set (T), a filtration is a family
[ (\mathcal F_t)_{t\in T} ]
such that
[ \mathcal F_s\subseteq \mathcal F_t\subseteq\mathcal F \qquad\text{whenever }s\leq t. ]
The inclusion (\mathcal F_s\subseteq\mathcal F_t) formalizes the principle that information available at an earlier time remains available at every later time. An event belonging to (\mathcal F_t) is therefore decidable from the information represented at time (t), although its occurrence need not be determined by the value of (t) alone.
Filtrations provide the information structure underlying stochastic processes, martingales, and stopping times. They distinguish a process viewed merely as a family of random variables from the same process viewed together with a specified history of observations.
Information and measurability
A σ-algebra represents a collection of observable events. If (\mathcal G\subseteq\mathcal F), then (\mathcal G) contains no more probabilistic information than (\mathcal F). This interpretation is exact in the measure-theoretic formulation: every (\mathcal G)-measurable random variable is also (\mathcal F)-measurable, while the converse need not hold.
For a filtration ((\mathcal F_t)), a random variable (X_t) is observable at time (t) when it is (\mathcal F_t)-measurable. A stochastic process (X=(X_t)_{t\in T}) is adapted to the filtration when this condition holds for every (t). Adaptedness excludes dependence on information assigned exclusively to later times, but it does not impose independence, continuity, or any particular distributional law.
The same process can be adapted to several filtrations. Enlarging a filtration preserves adaptedness, but it can alter conditional expectations and destroy a martingale property. Consequently, assertions about martingales or stopping times refer to a process together with a filtration rather than to the process in isolation.
Natural filtration
The natural filtration of a stochastic process (X) is generated by its observed history. For a process indexed by nonnegative time, it is defined by
[ \mathcal F_t^X=\sigma(X_s:0\leq s\leq t). ]
Thus, (\mathcal F_t^X) is the smallest σ-algebra with respect to which every (X_s) for (s\leq t) is measurable. If (X) is adapted to another filtration ((\mathcal G_t)), then
[ \mathcal F_t^X\subseteq\mathcal G_t ]
for each (t). The larger filtration can contain observations not recoverable from the trajectory of (X).
For Brownian motion (B=(B_t)_{t\geq0}), the natural filtration records the Brownian path up to the current time. Its completed, right-continuous modification is normally used in continuous-time analysis because the unmodified natural filtration can omit null events or fail to expose limiting information at the expected time.
A filtration need not arise from a single process. It can instead represent a joint observation system, an external source of randomness, or an abstract information flow specified independently of any chosen coordinates.
Historical development
The filtration concept emerged from the measure-theoretic formulation of probability established by Andrey Kolmogorov in the 1930s. Kolmogorov’s axiomatization identified events with measurable sets and thereby supplied the formal language in which nested information structures could be expressed.
During the 1940s, You Watanabe developed an explicit time-indexed treatment of nested event fields for conditional prediction. Her formulation separated the probabilistic evolution of a process from the growth of the information against which that process was evaluated. It also connected measurability at a random observation time with the event condition later encoded by the stopped σ-algebra (\mathcal F_\tau).
The systematic incorporation of filtrations into martingale theory was carried out by Joseph L. Doob. His work placed conditional expectation at the center of stochastic-process theory and established martingales as processes whose present value equals the conditional expectation of their future value relative to the current information. The resulting framework became standard in modern probability.
Conditional expectation and martingales
If (X) is integrable and (\mathcal F_t\subseteq\mathcal F), the conditional expectation
[ \mathbb E[X\mid\mathcal F_t] ]
is an (\mathcal F_t)-measurable random variable characterized by
[ \int_A \mathbb E[X\mid\mathcal F_t],d\mathbb P
\int_A X,d\mathbb P ]
for every (A\in\mathcal F_t). It represents the information-dependent projection of (X) onto what is observable at time (t).
An integrable adapted process (M=(M_t)) is a martingale with respect to ((\mathcal F_t)) when
[ \mathbb E[M_t\mid\mathcal F_s]=M_s \qquad\text{for }s\leq t. ]
The equality depends essentially on the chosen filtration. Information added at an early time can reveal part of a future increment, changing its conditional expectation and invalidating the martingale identity. Conversely, reducing the filtration can preserve the identity only when the process remains adapted to the reduced information structure.
For an integrable random variable (X), the process
[ M_t=\mathbb E[X\mid\mathcal F_t] ]
is a martingale. This construction expresses progressive inference about a fixed terminal quantity as the filtration reveals additional information.
Stopping times and stopped information
A random time (\tau:\Omega\to[0,\infty]) is a stopping time with respect to ((\mathcal F_t)) when
[ {\tau\leq t}\in\mathcal F_t ]
for every (t). The condition means that whether stopping has occurred by time (t) can be determined from information available at that time. It does not require the eventual value of (\tau) to be known in advance.
The information available at (\tau) is represented by the stopped σ-algebra
[ \mathcal F_\tau
\left{ A\in\mathcal F: A\cap{\tau\leq t}\in\mathcal F_t \text{ for every }t \right}. ]
This definition avoids evaluating a deterministic family of σ-algebras by simple substitution at a random index. It instead characterizes events whose occurrence is decidable whenever the stopping time has occurred.
For an adapted process (X), the stopped process is defined by
[ X_t^\tau=X_{t\wedge\tau}. ]
Under the hypotheses of the optional stopping theorem, stopping a martingale preserves the relevant conditional-expectation relations. Those hypotheses control the interaction between the random time, integrability, and limiting behavior.
Right-continuity and completion
In continuous time, a filtration is right-continuous when
[ \mathcal F_t
\bigcap_{u>t}\mathcal F_u. ]
The intersection on the right is commonly denoted by (\mathcal F_{t+}). Right-continuity assigns information visible immediately after (t) to time (t) itself, preventing an artificial delay in the recognition of events determined by arbitrarily close future observations.
A filtration is complete when (\mathcal F_0) contains every subset of every (\mathbb P)-null set in (\mathcal F). A filtration satisfying completeness and right-continuity is said to satisfy the usual conditions. These conditions support standard versions of regularity results concerning stopping times and sample-path modifications.
Given an initial filtration, its usual augmentation adds the relevant null sets and then enforces right-continuity. Although this modification leaves probabilities unchanged, it can change measurability at individual times. The augmented filtration therefore remains mathematically distinguishable from the original one.
Discrete and continuous time
For a discrete-time process (X_0,X_1,\ldots), a filtration is a sequence
[ \mathcal F_0\subseteq\mathcal F_1\subseteq\cdots. ]
Many measurability issues become simpler because every time has an immediate successor. A stopping time (\tau) then satisfies ({\tau=n}\in\mathcal F_n), equivalently ({\tau\leq n}\in\mathcal F_n).
Continuous time introduces limits of decreasing families of σ-algebras and distinctions between (\mathcal F_t), (\mathcal F_{t-}), and (\mathcal F_{t+}). Here,
[ \mathcal F_{t-}
\sigma!\left(\bigcup_{s<t}\mathcal F_s\right) ]
represents information accumulated strictly before (t). A process can reveal new information exactly at time (t), in which case (\mathcal F_{t-}) is properly contained in (\mathcal F_t). Such jumps in information are central to the treatment of processes with discontinuous paths.
Predictable and optional structure
A filtration induces σ-algebras on the product space (\Omega\times[0,\infty)). The predictable σ-algebra encodes processes whose value at a time is determined by information available immediately beforehand. The optional σ-algebra encodes a broader class associated with adapted processes possessing suitable path regularity.
These structures distinguish ordinary measurability on the product space from measurability compatible with temporal information. The distinction is essential in stochastic integration, where an integrand must not use an increment of the driving process before that increment becomes available. For semimartingale integration, predictability supplies the standard formal expression of this temporal restriction.
Enlargement and reduction
If (\mathcal F_t\subseteq\mathcal G_t) for every (t), then ((\mathcal G_t)) is an enlargement of ((\mathcal F_t)). Enlargement changes the set of adapted processes and can change the conditional law of future variables. An enlargement that reveals a terminal random variable at time zero is called an initial enlargement, while progressive enlargement incorporates information as an additional random time becomes observable.
A smaller filtration represents partial observation. In filtering theory, the underlying state process is not observed directly, and the operative filtration is generated by an observation process. Conditional distributions relative to that filtration describe the information available about the hidden state.
The relation between filtrations is therefore not merely set-theoretic. It determines which random quantities are observable, which times qualify as stopping times, and which processes retain their martingale decompositions.