Indistinguishable stochastic processes

Indistinguishable stochastic processes , including non-distinct stochastic processes called in are probability theory certain stochastic processes , the "small" only very negligible amounts and thus do not match each other. Indistinguishable processes can not be differentiated from one another using the given probability measure , since the “small” quantities have zero probability. The motivation for introducing indistinguishable stochastic processes is to examine the paths of stochastic processes, for example for continuity . These properties play an important role in the construction of more complex stochastic processes such as Brownian motion .

The modifications of a stochastic process are closely related and possibly identical to indistinguishability .

definition

Given are two stochastic processes and on the probability space with time set and state space . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle I}$ ${\displaystyle E}$

The processes and hot indistinguishable if there is a P-null set there, so that the amount of each in is included. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle N\in {\mathcal {A}}}$ ${\displaystyle \{X_{t}\neq Y_{t}\}}$ ${\displaystyle t\in I}$ ${\displaystyle N}$

properties

Indistinguishability of stochastic processes is a stronger term than that of the modifications of a stochastic process . This means that indistinguishable processes are always modifications of one another. Because according to the definition there is a zero set for each modification . In the case of indistinguishable processes, however, there is a zero set , so that . If such a zero set exists , the subsets of a zero set must all be zero sets. Conversely, if there are modifications of one another, it does not generally follow that the processes are also indistinguishable. This is because any union of the zero sets is generally no longer a zero set. ${\displaystyle X,Y}$ ${\displaystyle N_{t}=\{X_{t}\neq Y_{t}\}}$ ${\displaystyle t\in I}$ ${\displaystyle N}$ ${\displaystyle \textstyle \bigcup _{t\in I}N_{t}\subset N}$ ${\displaystyle N}$ ${\displaystyle N_{t}}$ ${\displaystyle X,Y}$ ${\displaystyle N_{t}}$

One example of this are the processes

${\displaystyle X_{t}={\begin{cases}1&{\text{ falls }}Z=t\\0&{\text{ falls }}Z\neq t\end{cases}}}$

such as

${\displaystyle Y_{t}=0}$.

Here is a normally distributed random variable. Then it's for everyone . So are and modifications of each other. But it can be shown that the processes are not indistinguishable. ${\displaystyle Z}$ ${\displaystyle P(X_{t}=Y_{t})=P(Z\neq t)=1}$ ${\displaystyle t\in I}$ ${\displaystyle X}$ ${\displaystyle Y}$

If there are modifications of a process with an index set (time set) , the reverse conclusion also applies under the following conditions, i.e. modifications of a process are indistinguishable. So the two terms are equivalent under the following circumstances: ${\displaystyle X,Y}$ ${\displaystyle I}$

• The index set is countable, because countable unions of zero sets are zero sets again ${\displaystyle I}$

or

• The processes and are almost certainly ${\displaystyle X}$ ${\displaystyle Y}$ continuous on the right-hand side and is an interval that can, however, be absolutely unlimited. ${\displaystyle I\subset \mathbb {R} }$

Individual evidence

1. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 270.

Web links

• AN Shiryaev: Stochastic indistinguishability . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 467-470 , 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. 269-270 , doi : 10.1007 / b137972 .