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''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

<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>

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>

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" .

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Dellacherie,   "Capacités et processus stochastiques" , Springer  (1972)</TD></TR></table>



====Comments====
For additional references, see [[Stochastic process|Stochastic process]].