Filtration¶
A filtration formalizes the gradual revelation of information over time. At time \(t\), the \(\sigma\)-algebra \(\mathcal{F}_t\) encodes exactly what can be decided from observations up to time \(t\). This structure underlies every notion that follows — adapted processes, martingales, stopping times, stochastic integrals.
σ-Algebras (Brief Recap)¶
A σ-algebra \(\mathcal{F}\) on \(\Omega\) is a collection of subsets closed under complements and countable unions, containing \(\Omega\). It is the collection of events to which probabilities can be assigned. For a collection \(\mathcal{C}\) of subsets, \(\sigma(\mathcal{C})\) denotes the smallest σ-algebra containing \(\mathcal{C}\). For a random variable \(X\),
is the σ-algebra of events decidable from observing \(X\). If \(\mathcal{F}\) is generated by \(k\) disjoint atoms, it contains exactly \(2^k\) sets.
Filtered Probability Spaces¶
A filtered probability space is \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})\), where \((\mathcal{F}_t)\) is an increasing family of sub-σ-algebras:
Interpretation
\(\mathcal{F}_t\) is the information available at time \(t\). Monotonicity says information is never forgotten — a family \((\mathcal{G}_t)\) with \(\mathcal{G}_s \not\subseteq \mathcal{G}_t\) for some \(s < t\) is not a filtration.
Extremes:
- Trivial: \(\mathcal{F}_t = \{\emptyset, \Omega\}\) for all \(t\) — nothing is ever revealed.
- Maximal: \(\mathcal{F}_t = \mathcal{F}\) for all \(t\) — everything is known from the start.
Terminal σ-algebra: \(\mathcal{F}_\infty := \sigma\!\left(\bigcup_{t \ge 0} \mathcal{F}_t\right)\) is the totality of information that eventually becomes available. (The union alone is typically only an algebra; see Exercise 2.)
Natural Filtrations¶
For a process \(X = (X_t)_{t \ge 0}\), the natural filtration is
the smallest filtration to which \(X\) is adapted (\(X_t\) is \(\mathcal{F}_t^X\)-measurable for every \(t\)).
Random walk filtration
Let \(X_1, X_2, \ldots\) be i.i.d. \(\pm 1\) fair flips with \(S_n = X_1 + \cdots + X_n\). The natural filtration is \(\mathcal{F}_n = \sigma(X_1, \ldots, X_n)\), generated by the \(2^n\) atoms corresponding to the possible first-\(n\)-flip sequences. An event is in \(\mathcal{F}_n\) iff its indicator is a function of \((X_1, \ldots, X_n)\): \(\{S_2 = 0\} \in \mathcal{F}_2\), but \(\{S_{10} > 5\} \notin \mathcal{F}_2\).
Brownian motion. For standard Brownian motion \(W\), the natural filtration \(\mathcal{F}_t^W = \sigma(W_s : s \le t)\) fails the usual conditions (below): it is not right-continuous (there exist immediate-future events like "BM returns to zero just after \(t\)" that live in \(\mathcal{F}_{t+}^W\) but not in \(\mathcal{F}_t^W\)), and \(\mathcal{F}_0^W = \{\emptyset, \Omega\}\) contains no non-trivial null sets.
The Usual Conditions¶
A filtration \((\mathcal{F}_t)\) satisfies the usual conditions if:
- Right-continuity: \(\mathcal{F}_t = \mathcal{F}_{t+} := \bigcap_{u > t} \mathcal{F}_u\) for all \(t \ge 0\).
- Completeness: \(\mathcal{F}_0\) contains all \(\mathbb{P}\)-null sets.
Right-continuity makes natural stopping times (hitting times such as \(\tau_a = \inf\{t : W_t > a\}\)) behave correctly — determining \(\{\tau_a \le t\}\) requires no "infinitesimal future" information. Completeness ensures the theory is insensitive to null-set modifications. Completeness of \(\mathcal{F}_0\) propagates to every \(\mathcal{F}_t\) by monotonicity.
Convention: Throughout stochastic calculus, we tacitly assume the usual conditions.
Augmentation¶
Given a natural filtration \((\mathcal{F}_t^W)\), the augmented filtration combines completion and right-continuification:
where \(\mathcal{N}\) is the collection of \(\mathbb{P}\)-null sets. The result satisfies the usual conditions and is the standard filtration used with Brownian motion.
Enlargement of Filtrations¶
Sometimes one needs a filtration larger than the natural one. Two standard constructions:
Initial enlargement adds a fixed random variable \(G\) from time zero:
Insider trading: if \(G = W_T\) (terminal stock price), then under \((\mathcal{G}_t)\) the insider knows \(W_T\) at every \(t\).
Progressive enlargement reveals a random time \(\tau\) as it occurs:
Credit default: before default we know only \(\{\tau > t\}\); at default we learn the exact value.
Warning
Enlargement can destroy the martingale property. \(W\) is a \((\mathcal{F}_t^W)\)-martingale but not a \((\mathcal{F}_t^W \vee \sigma(W_T))\)-martingale.
Summary¶
| Concept | Definition |
|---|---|
| Filtration | Increasing \((\mathcal{F}_t)\), \(\mathcal{F}_s \subseteq \mathcal{F}_t\) for \(s \le t\) |
| Natural filtration | \(\mathcal{F}_t^X = \sigma(X_s : s \le t)\) |
| Usual conditions | Right-continuous + complete |
| Augmentation | \(\bigcap_{u > t} \sigma(\mathcal{F}_u^W \cup \mathcal{N})\) |
| Initial enlargement | \(\mathcal{F}_t \vee \sigma(G)\) |
| Progressive enlargement | \(\mathcal{F}_t \vee \sigma(\mathbf{1}_{\{\tau \le t\}}, \tau \mathbf{1}_{\{\tau \le t\}})\) |
Filtrations are the backbone for everything that follows: conditional expectation (the next section) is always relative to a sub-σ-algebra, and martingales, stopping times, and stochastic integrals are always relative to a filtration.
Exercises¶
Exercise 1. Let \(X_1, X_2\) be i.i.d. fair \(\pm 1\) flips and \(S_n = X_1 + \cdots + X_n\). The σ-algebra \(\mathcal{F}_2 = \sigma(X_1, X_2)\) has 4 atoms.
(a) List the atoms and state \(|\mathcal{F}_2|\).
(b) For each of \(A = \{S_2 = 0\}\), \(B = \{S_1 > 0\}\), \(C = \{S_3 > 0\}\), \(D = \{\max_{k \le 2} S_k \ge 1\}\), decide whether it is \(\mathcal{F}_2\)-measurable.
(c) Compute \(\mathbb{E}[S_3 \mid \mathcal{F}_2]\).
Solution to Exercise 1
(a) Atoms: \(A_{++}, A_{+-}, A_{-+}, A_{--}\) according to the sign pattern of \((X_1, X_2)\). Then \(|\mathcal{F}_2| = 2^4 = 16\).
(b)
- \(A = \{S_2 = 0\} = A_{+-} \cup A_{-+}\) — measurable.
- \(B = \{S_1 > 0\} = A_{++} \cup A_{+-}\) — measurable.
- \(C = \{S_3 > 0\}\) depends on \(X_3\), independent of \(\mathcal{F}_2\) — not measurable.
- \(D = \{\max_{k \le 2} S_k \ge 1\} = A_{++} \cup A_{+-}\) (on \(A_{-\cdot}\) the max is \(\le 0\)) — measurable.
(c) \(\mathbb{E}[S_3 \mid \mathcal{F}_2] = S_2 + \mathbb{E}[X_3 \mid \mathcal{F}_2] = S_2 + \mathbb{E}[X_3] = S_2\), using independence of \(X_3\) from \(\mathcal{F}_2\).
Algebra vs σ-algebra
An algebra of sets is a collection closed under complements and finite unions, containing \(\Omega\). A σ-algebra is additionally closed under countable unions.
Every σ-algebra is an algebra, but not vice versa.
Exercise 2. Let \((\mathcal{F}_t)\) be a filtration.
(a) Show \(\mathcal{A} := \bigcup_{t \ge 0} \mathcal{F}_t\) is an algebra.
(b) Give a counterexample showing \(\mathcal{A}\) need not be a σ-algebra.
Solution to Exercise 2
(a) \(\Omega \in \mathcal{F}_0 \subseteq \mathcal{A}\). If \(A \in \mathcal{F}_t\) then \(A^c \in \mathcal{F}_t \subseteq \mathcal{A}\). If \(A \in \mathcal{F}_s, B \in \mathcal{F}_t\), then \(A, B \in \mathcal{F}_{\max(s,t)}\) (monotonicity), so \(A \cup B \in \mathcal{F}_{\max(s,t)} \subseteq \mathcal{A}\). \(\square\)
(b) Let \(\Omega = [0,1]\) and \(\mathcal{F}_n\) be generated by the dyadic partition of order \(n\). The atoms containing \(1/3\) shrink to \(\{1/3\}\), so \(\{1/3\} = \bigcap_n I_n\) with \(I_n \in \mathcal{F}_n \subseteq \mathcal{A}\). But \(\{1/3\} \notin \mathcal{F}_n\) for any finite \(n\) (every set in \(\mathcal{F}_n\) is a union of intervals of length \(2^{-n}\)). So \(\mathcal{A}\) is not closed under countable intersections. \(\square\)
Exercise 3. Prove: if \(\tau\) is a stopping time (i.e. \(\{\tau \le t\} \in \mathcal{F}_t\) for all \(t\)), then \(\{\tau < t\} \in \mathcal{F}_t\) for all \(t > 0\).
Solution to Exercise 3
\(\{\tau < t\} = \bigcup_{n=1}^\infty \{\tau \le t - 1/n\}\). Each \(\{\tau \le t - 1/n\} \in \mathcal{F}_{t - 1/n} \subseteq \mathcal{F}_t\) by monotonicity. Countable unions preserve \(\mathcal{F}_t\)-measurability. \(\square\)
Exercise 4. Let \(W\) be standard Brownian motion with natural filtration \((\mathcal{F}_t^W)\) and consider the initial enlargement \(\mathcal{G}_t = \mathcal{F}_t^W \vee \sigma(W_1)\).
(a) Is \(W_{1/2}\) \(\mathcal{G}_0\)-measurable? Is \(W_1\)?
(b) Compute \(\mathbb{E}[W_1 \mid \mathcal{G}_t]\) and show \(W\) is not a \((\mathcal{G}_t)\)-martingale.
Solution to Exercise 4
(a) \(\mathcal{G}_0 = \{\emptyset, \Omega\} \vee \sigma(W_1) = \sigma(W_1)\). So \(W_1\) is \(\mathcal{G}_0\)-measurable, but \(W_{1/2}\) is not: given \(W_1\), the value \(W_{1/2} = W_1/2 + B\) with \(B \sim N(0, 1/4)\) independent of \(W_1\) (Brownian bridge).
(b) Since \(\sigma(W_1) \subseteq \mathcal{G}_t\), \(W_1\) is \(\mathcal{G}_t\)-measurable. Taking out what is known: \(\mathbb{E}[W_1 \mid \mathcal{G}_t] = W_1\) for all \(t\). For a martingale we would need \(W_t = \mathbb{E}[W_1 \mid \mathcal{G}_t] = W_1\), which fails for \(t < 1\). \(\square\)
Exercise 5. Show that the trivial filtration \(\mathcal{F}_t = \{\emptyset, \Omega\}\) satisfies the usual conditions (assuming null sets are trivial). Then show that for any filtration \((\mathcal{F}_t)\), the right-continuification \(\mathcal{F}_t^+ := \bigcap_{s > t} \mathcal{F}_s\) is itself right-continuous.
Solution to Exercise 5
Trivial filtration: \(\bigcap_{s > t}\{\emptyset,\Omega\} = \{\emptyset,\Omega\} = \mathcal{F}_t\), and \(\emptyset \in \mathcal{F}_0\) covers the only null set. Usual conditions hold.
Right-continuification: We show \(\bigcap_{u > t} \mathcal{F}_u^+ = \mathcal{F}_t^+\).
- (\(\supseteq\)) For \(u > t\), since \(\{v : v > u\} \subset \{v : v > t\}\), intersecting over fewer sets yields a larger σ-algebra: \(\mathcal{F}_u^+ \supseteq \mathcal{F}_t^+\). Hence \(\bigcap_{u > t} \mathcal{F}_u^+ \supseteq \mathcal{F}_t^+\).
- (\(\subseteq\)) If \(A \in \bigcap_{u > t} \mathcal{F}_u^+\), then for every \(w > t\), picking any \(u \in (t, w)\) gives \(A \in \mathcal{F}_u^+ \subseteq \mathcal{F}_w\). So \(A \in \bigcap_{w > t} \mathcal{F}_w = \mathcal{F}_t^+\). \(\square\)