Filtration (abstract algebra)

From Academic Kids

In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with an index set I that is a totally ordered set, subject only to the condition that if ij in I then Si is contained in Sj. For example, in group theory, a filtration of a group G is a totally ordered set of subgroups of G, indexed in a particular way. The definition can be formulated as a monotone map from I to the set of subobjects.

Filtrations are widely used in abstract algebra, and homological algebra (where they are related in an important way to spectral sequences). In functional analysis other terminology is usually used, such as scale of spaces

In measure theory, in particular in martingale theory and the theory of stochastic processes, a filtration is a sequence of sigma-algebras on a measurable space. More formally, given a measurable space (Ω, F), a filtration is a sequence of sigma-algebras {Ft : 0 < t < ∞} with Ft contained in F for each t.

A sigma-algebra defines the set of events that can be measured, which in a probability context is equivalent to events that can be discriminated. Therefore a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of information. A typical example is in mathematical finance, where a filtration represents the information available at each time t, and is more and more precise (the set of measurable events is staying the same or increasing) as information from the present becomes available.

Therefore, a filtration is very often taken as strictly non-decreasing, i.e. for all s < t, Fs is contained in Ft.

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