Filtration (mathematics)

A filtration (including filtration, filtering or filters) is in the theory of stochastic processes, a family of nested σ - algebras. It models the information available at different times to the course of a random process.

Definition

Be an arbitrary ordered index set and a probability space.

A family of σ - algebras is called a filtration, if is sorted in ascending order, that is true with for all.

An alternative definition additional requirement that the family of σ - algebra is right-continuous, so that:

A stochastic process is called adapted to the filtration ( or adapted ) when always - measurable for all.

Called filtered probability space, when the sub - σ - algebras of.

Use of the term

The concept of filtering is essential to starting from the notion of the stochastic process, introduce important concepts such as martingales or stop times.

As a lot as in most stochastic processes or chosen and interpreted as time.

σ - algebras modeling the available information. The amounts of the σ - algebra to give at any time how much information currently known. For each event, translated means that at the time the question " is it? " Clearly "yes" or "no" can be answered. The fact that the filtration is always sorted in ascending order, therefore, means that, once obtained information not lost.

A stochastic process is a filtration adapted so this means that the behavior of the function in the time interval is well known ( for an arbitrary, but unknown and to be formulated with regard to the event question ).

The term is defined due to its importance in most advanced textbooks on stochastic processes. In some textbooks, for example the book by Albert N. Shiryaev Probability, the term is first introduced comprehensive didactic reasons for processes with discrete values ​​in discrete time.

Examples

Special Features

Fulfilled which belongs to the σ - algebra filtration terminal.

At any point is defined as the left limit of the filtration, the σ - algebra. Always according to. Votes filtration and left limit at any time agree, ie the filtration left -continuous. A continuous filtration is consequently one which is left - and right-continuous.