Adapted stochastic process

An adapted stochastic process is a special stochastic process in stochastics that fulfills certain measurability criteria. An adapted process can clearly remember the entire previous course of the process, i.e. it has all the information that has occurred up to that point in time . The availability of information is defined here via a filter . ${\displaystyle t}$ ${\displaystyle t}$

Adapted stochastic processes are central to the theory of the Martingale . Further stochastic processes that are defined using measurability criteria are the closely related predictable processes as well as the progressively measurable processes and the product-measurable processes that occur in the definition of the Ito integral .

definition

A probability space and a stochastic process with index set and values ​​in . Be a filtration in . ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle (X_{t})_{t\in T}}$ ${\displaystyle T}$ ${\displaystyle (E,{\mathcal {E}})}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}$ ${\displaystyle {\mathcal {A}}}$

Then a stochastic process is called -adapted or adapted to , if the following applies to each : ${\displaystyle \mathbb {F} }$ ${\displaystyle \mathbb {F} }$ ${\displaystyle t\in T}$

${\displaystyle X_{t}}$ is - - measurable . ${\displaystyle {\mathcal {F}}_{t}}$ ${\displaystyle {\mathcal {E}}}$

The index set can be any totally ordered set. In most cases, real-valued stochastic processes are considered, then is . ${\displaystyle T}$ ${\displaystyle (E,{\mathcal {E}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

Examples

If you choose to filter the complete information , that is

${\displaystyle {\mathcal {F}}_{t}:={\mathcal {A}}}$for all , ${\displaystyle t\in T}$

so every stochastic process is adapted with regard to this filtration. The measurability with regard to - already follows from the fact that each is a random variable . The measurability is then already included in the definition of the random variable. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle X_{t}}$

Define the filtration as

${\displaystyle {\mathcal {F}}_{t}:=\{\Omega ,\emptyset \}}$for , ${\displaystyle t\in T}$

thus σ-algebra the trivial σ-algebra, only a stochastic process is adapted, which consists of constant random variables. Because only constant functions - -measurable. Different random variables can, however, also assume different values, since this does not change the measurability. ${\displaystyle \{\Omega ,\emptyset \}}$ ${\displaystyle {\mathcal {E}}}$

Often a process is given its natural filtration

${\displaystyle {\mathcal {F}}_{t}:=\sigma (\{X_{s}\,|\,s\leq t\})}$.

By definition, it is the smallest filtration to which a given stochastic process is adapted.

Relationship to further measurability criteria

If a stochastic process is progressively measurable or product measurable , it is always adapted. This is based on the statement that a - -measurable function is still measurable with respect to when the second variable is fixed. Correspondingly, in the case of progressively measurable or product-measurable processes, a point in time is fixed and it is always - -measurable. ${\displaystyle {\mathcal {A}}_{1}\otimes {\mathcal {A}}_{2}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle t}$ ${\displaystyle X_{t}}$ ${\displaystyle {\mathcal {F}}_{t}}$ ${\displaystyle {\mathcal {E}}}$

Conversely, it can be shown: If an adapted stochastic process is continuous to the left or continuous to the right , it can be measured progressively.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .