Filtrations and Information

Motivation: why this matters in quant finance

A pricing model is not only about what can happen; it is about when information becomes available. At time

t

, a trader knows the history up to

t

and does not know the future. A trading rule that depends on

S_T

before time

T

is not a clever strategy; it is a model violation.

A filtration records this time structure. It is the object that makes phrases like "adapted process", "future Brownian increment", "stopping time", and "self-financing strategy" mathematically precise. Without it, the formula

\mathbb{E}^{\mathbb{Q}}[H\mid\mathcal{F}_t]

has no clear meaning because $\mathcal{F}_t$ has not been specified.

Bertsekas does not develop filtrations as a separate topic, but the ingredients come from Ch. 1's event collections and Ch. 4's conditional expectation as a forecast given information. This note assembles those ingredients into the dynamic framework used in stochastic finance.

The informal idea

A filtration is a growing list of questions the market can answer. At time $0$ , only initial data are known. At time $t$ , all events determined by the observed history up to $t$ are known. At time $T$ , more events are known.

The key word is growing. Information can be added but not erased:

\mathcal{F}_s\subseteq\mathcal{F}_t\qquad(s\le t).

Smaller sigma-algebras mean coarser information. Larger sigma-algebras mean finer information. This is why filtrations belong naturally after sigma-algebras and conditional expectation.

Formal definitions

Filtration

On a probability space

(\Omega,\mathcal{F},\mathbb{P})

, a filtration is a family

(\mathcal{F}_t)_{t\ge0}

of sub-sigma-algebras satisfying

\mathcal{F}_s\subseteq\mathcal{F}_t\subseteq\mathcal{F}\qquad\text{whenever }s\le t.

The quadruple

(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})

is a filtered probability space.

Natural filtration

For a process

(X_t)_{t\ge0}

, the natural filtration is

\mathcal{F}_t^X=\sigma(X_s:0\le s\le t).

It is the smallest filtration making the entire observed path up to time $t$ measurable.

Adapted process

A process

(X_t)

is adapted to

(\mathcal{F}_t)

X_t\text{ is }\mathcal{F}_t\text{-measurable for every }t.

The value at time $t$ can be determined from information available at time $t$ .

Stopping time

A random time

\tau

is a stopping time if

\{\tau\le t\}\in\mathcal{F}_t\qquad\text{for every }t.

At time $t$ , one can tell whether the time has already occurred.

Key properties

Information monotonicity

The defining inclusion $\mathcal{F}_s\subseteq\mathcal{F}_t$ says that every event knowable at $s$ remains knowable at $t$ . This is the mathematical version of remembering the past.

Adaptedness rules out clairvoyance

If a trading strategy $\Delta_t$ is adapted, then it cannot depend on $S_u$ for $u>t$ . This is the no-look-ahead condition underlying self-financing trading and Itô integration.

Future increments can be independent of current information

For Brownian motion, $W_t-W_s$ is independent of $\mathcal{F}_s$ for $s<t$ . This is stronger than saying it is uncorrelated with $W_s$ ; it says the whole past sigma-algebra carries no information about the future increment.

Conditional expectation depends on the filtration

A process can be a martingale under one filtration and not under another. The statement

\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s

is incomplete unless the filtration is specified.

Stopping times are non-anticipative random times

First hitting times are usually stopping times. Last hitting times before a terminal date usually are not, because deciding that a time is the last occurrence requires future information.

Worked examples

Example 1: two coin tosses

Let $\Omega=\{HH,HT,TH,TT\}$ . Define

\mathcal{F}_0=\{\emptyset,\Omega\},

\mathcal{F}_1=\{\emptyset,\{HH,HT\},\{TH,TT\},\Omega\},

and $\mathcal{F}_2=2^{\Omega}$ . At time $1$ you know the first toss but not the second. At time $2$ you know the full outcome. This is the simplest filtration that shows information becoming finer over time.

Example 2: a price process and adapted trading

If $S_t$ is the observed stock price, the natural market filtration is often modelled as $\mathcal{F}_t^S=\sigma(S_u:u\le t)$ . A delta hedge $\Delta_t=\Delta(t,S_t)$ is adapted because it uses current time and current price. A rule $\Delta_t=\Delta(t,S_T)$ is not adapted for $t<T$ and is excluded.

Example 3: barrier option exercise information

The first hitting time

\tau=\inf\{t:S_t\ge B\}

is a stopping time: by time $t$ , the observed path tells you whether the barrier has been hit. The last time before $T$ that $S_t\ge B$ is not a stopping time, because at time $t$ you do not know whether the path will cross again later.

Example 4: Brownian filtration

For Brownian motion $W$ , the natural filtration $\mathcal{F}_t^W$ contains every event determined by the path up to $t$ . The event $\{\sup_{0\le u\le t}W_u>B\}$ belongs to $\mathcal{F}_t^W$ . The event $\{W_{t+1}>B\}$ generally does not.

Common confusions and pitfalls

"The filtration is the realised history." The realised history is one path. The filtration is the collection of all events whose truth can be decided from histories up to each time.

"Adapted means deterministic." Adapted variables are known at the time in question, not known from the start.

"The natural filtration is always the market filtration." It is a common modelling choice, but not a law. In incomplete-information models, the market may observe less than the full state process.

"Every random time is a stopping time." A random time may depend on future information. Stopping times are precisely the random times that can be recognised without looking ahead.

"Changing the filtration is harmless." It can change martingale properties, admissible trading strategies, and no-arbitrage statements.

Where this goes next

Conditional Expectation: Uses $\mathcal{F}_t$ as the conditioning information.
Martingales Discrete Time: Defines fair games relative to a filtration.
Stopping Times: Develops the random-time concept in detail.
Optional Stopping Theorem: Studies martingales evaluated at stopping times.
Brownian Motion: Lives naturally on a filtered probability space.
Delta Hedging: Trading strategies must use only current and past market information.

References

Bertsekas, D. P., & Tsitsiklis, J. N. (2008). Introduction to Probability (2nd ed.). Athena Scientific. Ch. 1 §1.1-1.2 (events and probability laws), Ch. 4 §4.3 (conditional expectation as information-based forecasting). Bertsekas does not develop filtrations explicitly; this note extends those ingredients into the standard stochastic-finance framework.