Modifications of a stochastic process

Modifications of a stochastic process , also called versions of a stochastic process , are elements of certain equivalence classes of stochastic processes in probability theory . All stochastic processes that are very similar to one another in the sense that they cannot be differentiated at any point in time by the underlying probability measure are regarded as equivalent. Each of these processes is then a modification or version of a representative of this equivalence class. This classification is made in order to better examine the paths of stochastic processes. For example, the question of whether there is a modification of a stochastic process whose paths are continuous is interesting . This is important, for example, in the construction of Brownian motion . The Kolmogorov-Chentsov theorem makes a statement about the existence of locally Hölder-continuous modifications .

Closely related to the modifications of a stochastic process are the indistinguishable stochastic processes . Both terms may coincide.

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 are called modifications or versions of each other when that is almost certain for all . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle t\in I}$ ${\displaystyle X_{t}=Y_{t}}$

properties

The modifications of a stochastic process are a weaker term than indistinguishability . 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: ${\displaystyle X,Y}$ ${\displaystyle I}$

• Either the index set can be counted ${\displaystyle I}$
• 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

• AV Prokhorov: Stochastic equivalence . 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. 270 , doi : 10.1007 / b137972 .