1 What is a Filtration?
Consider a [[stochastic-process|stochastic process]] \((X_{t})\) defined on a probability space \((\Omega, \mathcal{F}, \mathbb{P})\).
A filtration \(\mathbb{F}:=(\mathcal{F}_{t})\) is a collection of \(\sigma\)-algebra that is increasing over time, that is \(\mathcal{F}_{s}\subseteq \mathcal{F}_{t}\) for all \(s \leq t\).
Intuitively, a filtration is the mathematical way of describing growing information as we move forwards in time. If the filtration contains information about a certain SP then we say that process is adapted.
We say that the process \((X_{t})\) is adapted to filtration \(\mathbb{F}\) if \(X_{t}\in \mathcal{F}_{t}\) for all \(t\).
That is, at time \(t\) we know \(X_{t}\) given \(\mathcal{F}_{t}\).
The natural filtration of a stochastic process is one which is generated from the process, e.g. \(\sigma((X_{t})_{t\in[0,T]})\), and can be understood intuitively as the information revealed specifically about the stochastic process up to present time \(T\).
In most rigorous treatments, particularly for continuous-time processes, filtrations are assumed to satisfy the usual conditions:
- Completeness: \(\mathcal{F}_{0}\) contains all \(\mathbb{P}\)-null sets
- Right-continuity: \(\mathcal{F}_{t} = \bigcap_{s>t} \mathcal{F}_{s}\) (filtration is right-continuous in time)
These conditions ensure nice properties for processes and integrals, though they’re sometimes relaxed depending on context.
2 Examples
Adapted process: Brownian Motion \((B_{t})\) with its natural filtration \(\mathcal{F}_{t} = \sigma(B_{s} : s \leq t)\) is adapted because you know \(B_{t}\) at time \(t\). The information revealed so far includes all past and present values of the path.
Non-adapted process: If \(X_{t}\) depends on a future value like \(X_{t} = B_{t+1}\), it is not adapted to \(\mathcal{F}_{t}\) because you don’t have information about \(B_{t+1}\) at time \(t\) (you cannot know the future).
3 Why This Matters
- Martingale Theory: Adapted processes are essential—martingales are defined with respect to a filtration and must be adapted to meaningfully capture “fair games.”
- Predictability & no-arbitrage: Adaptedness ensures processes only depend on current and past information, not future information. This is fundamental to no-arbitrage conditions in finance.
- Stochastic Calculus: The Itô integral requires the integrand to be adapted, you cannot use future information in trading strategies.