Optional sigma-algebra
optional -
algebra
The smallest \sigma - algebra {\mathcal O} = {\mathcal O} ( \mathbf F ) of sets (cf. Algebra of sets) in \Omega \times \mathbf R _ {+} = \{ {( \omega , t) } : {\omega \in \Omega, t \geq 0 } \} generated by all mappings ( \omega , t) \rightarrow f( \omega , t) of the set \Omega \times \mathbf R _ {+} into \mathbf R which (for every fixed \omega \in \Omega ) are continuous from the right (in t ), have limits from the left and are adapted to a (given) non-decreasing family \mathbf F = ( F _ {t} ) _ {t \geq 0 } of sub- \sigma - algebras F _ {t} \subseteq F , t \geq 0 , where ( \Omega , F ) is a measurable space. The optional \sigma - algebra coincides with the smallest \sigma - algebra generated by the stochastic intervals [ 0, \tau ] = \{ {( \omega , t) } : {0 \leq t < \tau ( \omega ) } \} , where \tau = \tau ( \omega ) are stopping times (relative to \mathbf F = ( F _ {t} ) _ {t \geq 0 } ) (cf. Markov moment). The inclusion {\mathcal P} ( \mathbf F ) \subseteq {\mathcal O} ( \mathbf F ) holds between the optional and predictable \sigma - algebras (cf. Predictable sigma-algebra).
References
[1] | C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) |
Comments
In [a1] the optional \sigma - field is called the well-measurable \sigma - field.
References
[a1] | C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A , North-Holland (1978) (Translated from French) |
Optional sigma-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Optional_sigma-algebra&oldid=48061