Probability-2 by Albert N. Shiryaev

Author:Albert N. Shiryaev
Language: eng
Format: epub
ISBN: 9780387722085
Publisher: Springer New York

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It is natural to say, by analogy with Definition 1 in Sect. 1, that the process Z = (Z t)t≥0 is a martingale (with respect to the “flow” of σ-algebras). Notice that in this case , t ≥ 0 (cf. (1) in Sect. 1).

By analogy with Definition 3 in Sect. 1, we shall say that the random variable τ = τ(ω) with values in [0, +∞] is a Markov time, or a random variable independent of the future (relative to the “flow” of σ-algebras ) if for each t ≥ 0 the set

It turns out that for martingales with continuous time, which are considered now, Theorem 1 from Sect. 2 remains valid (with self-evident changes to the notation). In particular,

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