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Predictable sigma-algebra

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predictable -algebra

The least -algebra of sets in

generated by all mappings of the set into that are (for each fixed ) continuous from the left (in ) and -adapted to a non-decreasing family of sub--algebras , , where is a measurable space. A predictable -algebra can be generated by any of the following families of sets:

1) , where and , where is a stopping time (cf. Markov moment) and a stochastic interval;

2) , where , and , where and .

Between optional -algebras (cf. Optional sigma-algebra) and predictable -algebras there is the relation .

References

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


Comments

Instead of "(s-) algebra" one more often uses (-) field.

References

[a1] C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A-C , North-Holland (1978–1988) (Translated from French)
How to Cite This Entry:
Predictable sigma-algebra. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Predictable_sigma-algebra&oldid=39353
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article