1 Filtration
A filtration \(\{ \mathcal{F}_{n},n \geq 1 \}\) is a family of \(\sigma\)-algebra that is increasing in time \(\mathcal{F}_{n}\subseteq \mathcal{F}_{n+1}\) for \(n \geq 1\).
The most important type of filtration are those generated or adapted to stochastic processes. For example, if there is a sequence of random variables \(X_{n},~n\geq 1\) and we define \(\mathcal{F}_{n}:=\sigma(X_{1}, \dots, X_{n})\) to be the \(\sigma\)-algebra generated by \(\{ X_{n}, n\geq 1 \}\), then \(\{ \mathcal{F}_{n},n\geq 1 \}\) forms a filtration and we call it the filtration generated by the sequence \(\{ X_{n}, n\geq 1 \}\).
We think of such a filtration as an increasing information set for the stochastic process \(\{ X_{n},n\geq 1 \}\).