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We study Doob's martingale convergence theorem for computable continuous timemartingales on Brownian motion, in the context of algorithmic randomness. Acharacterization of the class of sample points for which the theorem holds isgiven. Such points are given the name of Doob random points. It is shown that apoint is Doob random if its tail is computably random in a certain sense.Moreover, Doob randomness is strictly weaker than computable randomness and isincomparable with Schnorr randomness.

Algorithmic randomness for Doob's martingale convergence theorem in continuous time