Filtration (probability theory, $({\mathcal{F}}_t)$)

A filtration is an increasing family of sigma-algebras: ${\mathcal{F}}_s\subseteq {\mathcal{F}}_t$ whenever $s\le t$. It represents the information available as time passes.

On $(0,1{]}^d$ with Lebesgue probability measure, the dyadic filtration is generated by the partitions into half-open cubes of side $2^{-n}$. The conditional expectation of an integrable function is its average on each atom. The increasing partitions generate the Borel sigma-algebra. Step functions on these partitions are dense in $L^1$; approximation and the Doob maximal inequality show that their conditional averages converge in $L^1$ and almost everywhere to the original function.

On $(0,1]$ with Lebesgue probability and the usual dyadic filtration, each displayed support is a dyadic atom. Its next split has one child of doubled height and one of zero height, preserving the conditional average. Hence the sequence is a martingale from time one. To extend it to the trivial time-zero sigma-algebra one must set $X_0=1$: applying the same displayed formula at zero instead gives $1_{(0,1/2]}$, which is not adapted and has the wrong expected value. The repaired martingale tends to zero almost surely while its expected values remain one, so it is not uniformly integrable.

A Filtered probability space is a probability space together with an increasing family of sigma-algebras ${\mathcal{F}}_t\subseteq \mathcal{F}$ describing the information available by time $t$.

The usual conditions are completeness and right continuity of a filtration: ${\mathcal{F}}_0$ contains all subsets of null events in the ambient probability space, and ${\mathcal{F}}_t={\bigcap }_{s>t}{\mathcal{F}}_s$. They allow the standard continuous-time martingale, stopping time and stochastic integration theorems to be used without repeated augmentation qualifications. An absolute continuity of measures change preserves old null sets but can introduce new ones; the usual completion under the new measure may therefore be understood when needed.

For a filtration $({\mathcal{F}}_t)$ and $t>0$, the information strictly before time $t$ isA predictable process evaluated at a deterministic positive time $t$ is ${\mathcal{F}}_{t-}$-measurable.