Optional sigma-algebra

optional $ \sigma $- 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)