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Difference between revisions of "Optional random process"

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A [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068560/o0685601.png" /> that is measurable (as a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068560/o0685602.png" />) with respect to the [[Optional sigma-algebra|optional sigma-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068560/o0685603.png" />.
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A [[stochastic process]] $X = (X_t(\omega),F_t)_{t\ge0}$ that is [[measurable mapping|measurable]] (as a mapping $(\omega,t) \mapsto X(\omega,t) = X_t(\omega)$) with respect to the [[optional sigma-algebra]] $\mathcal{O} = \mathcal{O}(\mathbf{F})$.
  
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,  "Capacités et processus stochastiques" , Springer  (1972)  pp. Chapt. 3, Sect. 2</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bauer,  "Probability theory and elements of measure theory" , Holt, Rinehart &amp; Winston  (1972)  pp. Chapt. 11  (Translated from German)</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Dellacherie,  "Capacités et processus stochastiques" , Springer  (1972)  pp. Chapt. 3, Sect. 2</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bauer,  "Probability theory and elements of measure theory" , Holt, Rinehart &amp; Winston  (1972)  pp. Chapt. 11  (Translated from German)</TD></TR>
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Latest revision as of 20:27, 1 October 2016

A stochastic process $X = (X_t(\omega),F_t)_{t\ge0}$ that is measurable (as a mapping $(\omega,t) \mapsto X(\omega,t) = X_t(\omega)$) with respect to the optional sigma-algebra $\mathcal{O} = \mathcal{O}(\mathbf{F})$.


Comments

An optional random process is also called an adapted random process.

References

[a1] C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) pp. Chapt. 3, Sect. 2
[a2] H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) pp. Chapt. 11 (Translated from German)
How to Cite This Entry:
Optional random process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Optional_random_process&oldid=12518
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article