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Optional sigma-algebra

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optional -algebra

The smallest -algebra of sets (cf. Algebra of sets) in generated by all mappings of the set into which (for every fixed ) are continuous from the right (in ), have limits from the left and are adapted to a (given) non-decreasing family of sub--algebras , , where is a measurable space. The optional -algebra coincides with the smallest -algebra generated by the stochastic intervals , where are stopping times (relative to ) (cf. Markov moment). The inclusion holds between the optional and predictable -algebras (cf. Predictable sigma-algebra).

References

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


Comments

In [a1] the optional -field is called the well-measurable -field.

References

[a1] C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A , North-Holland (1978) (Translated from French)
How to Cite This Entry:
Optional sigma-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Optional_sigma-algebra&oldid=48061
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article