Filtration (mathematics)

In mathematics, a filtration ${\displaystyle {\mathcal {F}}}$ is an indexed set S_i of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that

If the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic object S_i gaining in complexity with time. Hence, a process that is adapted to a filtration ${\displaystyle {\mathcal {F}}}$, is also called non-anticipating, i.e. one that cannot see into the future.

Sometimes, as in a filtered algebra, there is instead the requirement that the ${\displaystyle S_{i}}$ be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication), that satisfy ${\displaystyle S_{i}\cdot S_{j}\subset S_{i+j}}$, where the index set is the natural numbers; this is by analogy with a graded algebra.

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