Predictable sigma-algebra
From Encyclopedia of Mathematics
predictable -algebra
The least -algebra
of sets in
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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://encyclopediaofmath.org/index.php?title=Predictable_sigma-algebra&oldid=48279
Predictable sigma-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Predictable_sigma-algebra&oldid=48279
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article