# Optional random process

From Encyclopedia of Mathematics

A stochastic process $X = (X_t(\omega),F_t)_{t\ge0}$ that is measurable (as a mapping $(\omega,t) \mapsto X(\omega,t) = X_t(\omega)$) with respect to the optional sigma-algebra $\mathcal{O} = \mathcal{O}(\mathbf{F})$.

#### Comments

An optional random process is also called an adapted random process.

#### References

[a1] | C. Dellacherie, "Capacités et processus stochastiques" , Springer (1972) pp. Chapt. 3, Sect. 2 |

[a2] | H. Bauer, "Probability theory and elements of measure theory" , Holt, Rinehart & Winston (1972) pp. Chapt. 11 (Translated from German) |

**How to Cite This Entry:**

Optional random process.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Optional_random_process&oldid=39349

This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article