Probability happens in time. As the clock ticks, outcomes get revealed and we learn more.
A filtration is the mathematical record of what we know by time
t: an increasing family of
σ-algebras
(\mathcal{F}_t)_{t \ge 0} with
\mathcal{F}_s \subseteq \mathcal{F}_t \qquad \text{whenever } s \le t.
The containment is the whole point: information only ever accumulates. Every
event we could resolve by time s we can still resolve later at
t — nothing is forgotten. Bigger
\sigma-algebra, finer questions answerable, more known. Think of
\mathcal{F}_t as the list of yes/no questions whose answer is settled by
time t: as the clock advances, questions only ever move onto the
list, never off it. A filtration is therefore the formal model of memory that is never erased.
The natural filtration of a process
Where does a filtration come from? Usually from the very process we are watching. Given a stochastic
process (X_t)_{t \ge 0}, the most economical answer to "what is known by
time t?" is "everything the path of X has
revealed so far". That is the natural filtration of
X:
\mathcal{F}_t = \sigma\big(X_s : s \le t\big),
the \sigma-algebra generated by all the observations
X_s up to and including time t. It is the
smallest \sigma-algebra making every one of those random variables
measurable — exactly the information you would hold if you had watched
X from the start of time until now and nothing else.
We claim this family is a genuine filtration, i.e. it is increasing:
\mathcal{F}_s \subseteq \mathcal{F}_t whenever
s \le t. Here is the derivation, one step at a time.
Step 1 — compare the generating families. The two
\sigma-algebras are generated by the two collections of observations
\mathcal{C}_s = \{\,X_u : u \le s\,\}, \qquad \mathcal{C}_t = \{\,X_u : u \le t\,\}.
Fix s \le t. If u \le s then automatically
u \le t (transitivity of \le), so every
observation in the smaller family also appears in the larger one:
\{\,X_u : u \le s\,\} \subseteq \{\,X_u : u \le t\,\}, \qquad \text{i.e.}\quad \mathcal{C}_s \subseteq \mathcal{C}_t.
Step 2 — generation is monotone. The map
\mathcal{C} \mapsto \sigma(\mathcal{C}) is order-preserving: a larger
family of generators can only produce a larger
\sigma-algebra. Concretely, \sigma(\mathcal{C}_t)
is a \sigma-algebra that makes every member of
\mathcal{C}_t measurable, and since
\mathcal{C}_s \subseteq \mathcal{C}_t it makes every member of the
sub-collection \mathcal{C}_s measurable too. But
\sigma(\mathcal{C}_s) is by definition the smallest
\sigma-algebra doing that, so it must be contained in any other one that
also does — in particular in \sigma(\mathcal{C}_t):
\mathcal{C}_s \subseteq \mathcal{C}_t \;\Longrightarrow\; \sigma(\mathcal{C}_s) \subseteq \sigma(\mathcal{C}_t).
Step 3 — read off the conclusion. Substituting the definitions
\mathcal{F}_s = \sigma(\mathcal{C}_s) and
\mathcal{F}_t = \sigma(\mathcal{C}_t) gives exactly the filtration property:
\mathcal{F}_s \subseteq \mathcal{F}_t \qquad \text{whenever } s \le t.
So the natural filtration is increasing for free — it is built that way, because "the questions
answerable from a longer record" can only be more numerous than those from a shorter one.
For any process (X_t)_{t \ge 0}, its natural filtration
\mathcal{F}_t = \sigma(X_s : s \le t) satisfies
\mathcal{F}_s \subseteq \mathcal{F}_t for all
s \le t, and so is a filtration. The reason is monotonicity of
generation: s \le t gives
\{X_u : u \le s\} \subseteq \{X_u : u \le t\}, and a larger family of
generators yields a larger \sigma-algebra,
\sigma(\cdot) being monotone.
The refining tree
Picture tossing a coin at each time step. At t = 0 you know nothing:
\mathcal{F}_0 = \{\varnothing, \Omega\} is trivial — you cannot
distinguish any outcome from any other. The first toss splits the world in two; the second splits
each half again. Each \mathcal{F}_t is the
\sigma-algebra generated by the partition into these blocks, and the
partitions get strictly finer:
\mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 \subset \cdots
Step through the tree and watch the partition refine — each new layer is a larger
\sigma-algebra sitting on top of the last.
Adapted processes
A stochastic process (X_t) is adapted to the filtration
if each X_t is \mathcal{F}_t-measurable —
its value is known by time t. Adaptedness rules out clairvoyance:
you may use only information already revealed, never a peek at the future.
Spelled out, adaptedness is the requirement
X_t \text{ is } \mathcal{F}_t\text{-measurable for every } t \ge 0.
Equivalently, every event of the form \{X_t \in B\} (for a
Borel set
B) lies in \mathcal{F}_t — its truth can be
decided from information available by time t. A process is automatically
adapted to its own natural filtration; the content of the condition appears when
(\mathcal{F}_t) is some larger, externally specified flow of information
and we insist X not run ahead of it.
This is the backbone of everything that follows. A
martingale
is an adapted process whose
conditional expectation
of tomorrow, given today's \mathcal{F}_t, is today's value — and the Itô
integral is built so its integrand is adapted, never anticipating the Brownian increment it is paid
against.
Adaptedness says X_t is known at time
t. A subtly stronger notion is
predictability: X_t is known just before
time t. In discrete time the distinction is crisp — a process
(H_n) is predictable when H_n is
\mathcal{F}_{n-1}-measurable, i.e. fixed by the information one step
earlier:
H_n \text{ is } \mathcal{F}_{n-1}\text{-measurable}, \qquad \text{whereas adapted only asks } \mathcal{F}_{n}\text{-measurable}.
Why care? Predictability is exactly the honesty condition for a trading strategy. You must
choose the size of your bet on step n before the
n-th outcome is revealed — using \mathcal{F}_{n-1}
only. A merely adapted strategy could peek at the very increment it is betting on, which is
tantamount to insider trading. We will see this again in the martingale transform, where the
predictability of the strategy is precisely what keeps a fair game fair.
In continuous time, textbooks routinely demand that the filtered probability space satisfy the
usual conditions. These are two technical hypotheses imposed on
(\mathcal{F}_t) to make the theory run smoothly:
-
Right-continuity:
\mathcal{F}_t = \mathcal{F}_{t^+} := \bigcap_{s > t} \mathcal{F}_s.
Nothing new is learned in the instant strictly after t that was not
already available at t. This rules out information arriving
"infinitesimally late" and is what lets hitting times such as
\tau = \inf\{t : X_t > a\} be genuine
stopping times.
-
Completeness: \mathcal{F}_0 (hence every
\mathcal{F}_t) contains all the
\mathbb{P}-null sets of \mathcal{F}. So if
two processes agree almost surely, the filtration cannot tell them apart — a measurability
statement proved for one transfers automatically to any version differing on a null set.
Both are imposed so that "almost sure" reasoning, continuous-time path properties, and stopping-time
arguments don't break on measure-zero pathologies. Any natural filtration can be enlarged to satisfy
them (its usual augmentation), and one quietly assumes this has been done.