Predictable sigma-algebra
From Encyclopedia of Mathematics
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://encyclopediaofmath.org/index.php?title=Predictable_sigma-algebra&oldid=16866
Predictable sigma-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Predictable_sigma-algebra&oldid=16866
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article