Filtration (probability theory)

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A filtration (also filtration , filtering or filtering ) is a family of nested σ-algebras in the theory of stochastic processes . It models the information available at different points in time on the course of a random process.

definition

Let be an index set and a probability space .

Furthermore, let a sub-σ-algebra of be given for each .

Then it is called the family of σ-algebras

a filtration or filtration (in or on ), if it is arranged in ascending order, that is:

For everyone with applies .

If there is a filtration, then a filtered probability space is also called.

Similarly, filtering can also be defined for any semi-ordered index quantities.

example

If one considers as an example a probability space with a countable basic set , which is equipped as standard with the power set as σ-algebra, then a possible filtering would be for example

.

It models the information that up to the nth time step one has moved up to n steps away from the origin and would, for example, be the suitable filter for a simple symmetrical random walk .

Special filterings

Filtration produced

If a stochastic process, the system created by it is called the created filtration , canonical filtration or natural filtration of the process ( denotes the σ-algebra operator ). Complete information about the past course of the process up to and including the point in time is therefore available at all times .

Filtration of the complete information

The definition for all defines the filtering of the complete information . The complete information is available here at all times .

Continuous filterings

Defined for a filtration

as well as and , so applies

.

Is

  • , the filtration is called a right- hand continuous filtration or right-hand continuous,
  • , so the filtration is called a left- hand continuous filtration or left-hand continuous,
  • Continuous on the left and on the right, one speaks of continuous filtration .

Filtration of stop times

A stop time with regard to any filtration generates, in analogy to natural filtration, a σ-algebra, the so-called σ-algebra of the τ past

with .

Let there be an ordered family of stop times with for everyone with , then the family is a filtration, this is important when studying stop times of stochastic processes. The continuous right-hand version of the filtration is generated analogously , where:

and .

It always applies .

Augmented filtration

An augmented filtration is the equivalent of completing a measurement space for filtrations. If there is a probability space and a filtration, then one defines

as a set system of (not necessarily measurable) subsets of zero sets . The augmented filtration (of relative ) is then defined as

and

.

Standard filtration and the usual conditions

A filtration is called a standard filtration if it agrees with your augmented filtration and is steady to the right, i.e. if

applies. It is then also said that the usual conditions apply.

You can change from any filtration to a standard filtration by first going to the steady-right and then to the augmented filtration.

Use of the term

The concept of filtration is essential in order to introduce important concepts such as martingales or stop times , based on the concept of the stochastic process .

As with stochastic processes, or is usually chosen as the quantity and interpreted as the point in time.

σ-algebras model available information. The sets of σ-algebra indicate how much information is currently known at any point in time. Translated for every event, it means that the question “is ?” Can be clearly answered with “yes” or “no” at the time. The fact that the filtration is always in ascending order means that information that has been obtained is no longer lost.

If a stochastic process is adapted to a filtration , this means that the course of the function in the interval at the point in time is known (for any, but unknown and with regard to the questions that can be formulated by events ).

The term is defined by its meaning in most advanced textbooks on stochastic processes. In some textbooks, for example in the book Probability by Albert N. Schirjajew , the term is initially introduced comprehensively for processes with discrete values ​​in discrete time for didactic reasons.

literature

Individual evidence

  1. Klenke: Probability Theory. 2013, p. 195.
  2. Meintrup, Schäffler: Stochastics. 2005, p. 390.
  3. Meintrup, Schäffler: Stochastics. 2005, p. 390.
  4. Klenke: Probability Theory. 2013, p. 482.