Optional random process

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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 $\mathcal{O} = \mathcal{O}(\mathbf{F})$.


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


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