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Difference between revisions of "Stochastic process, compatible"

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''adapted stochastic process''
 
''adapted stochastic process''
  
A family of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s0902001.png" /> defined on a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s0902002.png" />, with an increasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s0902003.png" /> of sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s0902004.png" />-fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s0902005.png" />, such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s0902006.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s0902007.png" />-measurable for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s0902008.png" />. In order to stress this property for such processes, one often uses the notation
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A family of random variables $X=(X_t(\omega))_{t\geq0}$ defined on a measurable space $(\Omega,\mathcal F)$, with an increasing family $\mathbf F=(\mathcal F_t)_{t\geq0}$ of sub-$\sigma$-fields $\mathcal F_t\subseteq\mathcal F$, such that the $X_t$ are $\mathcal F_t$-measurable for every $t\geq0$. In order to stress this property for such processes, one often uses the notation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s0902009.png" /></td> </tr></table>
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$$X=(X_t,\mathcal F_t)_{t\geq0}$$
  
 
or
 
or
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s09020010.png" /></td> </tr></table>
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$$X=(X_t,\mathcal F_t),$$
  
and says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s09020011.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s09020013.png" />-adapted, or adapted to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s09020014.png" />, or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090200/s09020015.png" /> is an adapted process. Corresponding definitions can also be given in the case of discrete time, and then  "adapted process"  is sometimes replaced by  "adapted sequence" .
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and says that $X$ is $\mathbf F$-adapted, or adapted to the family $\mathbf F=(\mathcal F_t)_{t\geq0}$, or that $X$ is an adapted process. Corresponding definitions can also be given in the case of discrete time, and then  "adapted process"  is sometimes replaced by  "adapted sequence" .
  
 
====References====
 
====References====

Latest revision as of 12:10, 26 July 2014

adapted stochastic process

A family of random variables $X=(X_t(\omega))_{t\geq0}$ defined on a measurable space $(\Omega,\mathcal F)$, with an increasing family $\mathbf F=(\mathcal F_t)_{t\geq0}$ of sub-$\sigma$-fields $\mathcal F_t\subseteq\mathcal F$, such that the $X_t$ are $\mathcal F_t$-measurable for every $t\geq0$. In order to stress this property for such processes, one often uses the notation

$$X=(X_t,\mathcal F_t)_{t\geq0}$$

or

$$X=(X_t,\mathcal F_t),$$

and says that $X$ is $\mathbf F$-adapted, or adapted to the family $\mathbf F=(\mathcal F_t)_{t\geq0}$, or that $X$ is an adapted process. Corresponding definitions can also be given in the case of discrete time, and then "adapted process" is sometimes replaced by "adapted sequence" .

References

[1] C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972)


Comments

For additional references, see Stochastic process.

How to Cite This Entry:
Stochastic process, compatible. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_process,_compatible&oldid=32548
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article