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

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.

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:

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.

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.