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Filtration (mathematics)


In mathematics, a filtration is an indexed set Si 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 Si gaining in complexity with time. Hence, a process that is adapted to a filtration , 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 be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication), that satisfy , where the index set is the natural numbers; this is by analogy with a graded algebra.


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