Predictable sigma-algebra

predictable $ \sigma $- algebra

The least $ \sigma $- algebra $ {\mathcal P} = {\mathcal P} ( \mathbf F ) $ 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 $ that are (for each fixed $ \omega \in \Omega $) continuous from the left (in $ t $) and $ \mathbf F $- adapted to a non-decreasing family $ \mathbf F = ( {\mathcal F} _ {t} ) _ {t \geq 0 } $ of sub- $ \sigma $- algebras $ {\mathcal F} _ {t} \subseteq {\mathcal F} $, $ t \geq 0 $, where $ ( \Omega , {\mathcal F} ) $ is a measurable space. A predictable $ \sigma $- algebra can be generated by any of the following families of sets:

1) $ A \times \{ 0 \} $, where $ A \in {\mathcal F} _ {0} $ and $ [[ 0, \tau ]] $, where $ \tau $ is a stopping time (cf. Markov moment) and $ [[ 0, \tau ]] $ a stochastic interval;

2) $ A \times \{ 0 \} $, where $ A \in {\mathcal F} _ {0} $, and $ A \times ( s, t] $, where $ s < t $ and $ A \in {\mathcal F} _ {s} $.

Between optional $ \sigma $- algebras (cf. Optional sigma-algebra) and predictable $ \sigma $- algebras there is the relation $ {\mathcal P} ( \mathbf F ) \subseteq {\mathcal O} ( \mathbf F ) $.

References

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

Comments

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

References

[a1]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A-C , North-Holland (1978–1988) (Translated from French)