Path (stochastics)

In stochastics, the realizations of a stochastic process are called a path . If the index set of the process is interpreted as time and the values ​​of the process as spatial position, then the process "runs" along a path with increasing time. It is important here that the paths are concretizations or evaluations of the stochastic process. This can be imagined as follows: The stochastic process has a certain potential to assume certain states, just as a cube has a potential to show a certain number of points. A path of a process now corresponds to a concretization of this potential, using the example of the dice this corresponds to the determination of a number by throwing the dice.

definition

Two example paths of a standard Wiener process

Given is a stochastic process on the probability space with an index set that takes values ​​in . ${\displaystyle X=(X_{t})_{t\in I}}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle E}$

Then means for the figure ${\displaystyle \omega \in \Omega }$ ${\displaystyle t\mapsto X_{t}(\omega )}$

with definition set and target set a path of . ${\displaystyle I}$ ${\displaystyle E}$ ${\displaystyle X}$

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Often one is interested in the properties of paths such as continuity . To do this, you need an additional structure on the target set, such as a metric . In general, paths are functions in a real variable; with a corresponding structure on the target set, paths can therefore also have properties such as differentiability or the like. ${\displaystyle E}$

If the paths have the property for almost all of them , it is also said that the process almost certainly has the property . Thus, one can also meaningfully speak of continuous, càdlàg or differentiable stochastic processes. ${\displaystyle \omega \in \Omega }$ ${\displaystyle t\mapsto X_{t}(\omega )}$ ${\displaystyle A}$ ${\displaystyle A}$

example

As an example consider an independently identical Bernoulli-distributed sequence of random variables with parameters . This corresponds to a Bernoulli process . One possible path would be ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle p\in (0,1)}$

${\displaystyle 0-0-1-1-0-1-1-1-0-0-1-0-1-\dots }$,

another

${\displaystyle 1-0-0-1-0-1-0-1-0-1-1-0-1-\dots }$.

The concrete probability of the path is irrelevant here as long as it is possible.

Use: visualization and classification of stochastic processes

Paths are often used in the modeling and visualization of stochastic processes. In addition, some stochastic processes are defined by the properties of their paths. That's what a stochastic process is called one

• left- hand continuous process when almost all left-hand paths are continuous
• fairly steady process , when almost all the paths right continuous are
• continuous process , when almost all paths constantly are
• RCLL process or càdlàg process , when almost all paths RCLL are
• differentiable, Hölder continuous etc., if almost all paths are differentiable , Hölder continuous etc.

The definition of the Wiener process, for example, requires that it should be continuous.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 469 , 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 , pp. 269 , doi : 10.1007 / b137972 .