let "a" be an event related to the wiener process (more formally: a set, measurable with respect to the wiener measure, in the space of functions), and "x" the conditional probability of "a" given the wiener process on the time interval [0, "t"] (more formally: the wiener measure of the set of trajectories whose concatenation with the given partial trajectory on [0, "t"] belongs to "a"). then the process "x" is a continuous martingale. its martingale property follows immediately from the definitions, but its continuity is a very special fact – a special case of a general theorem stating that all brownian martingales are continuous. a brownian martingale is, by definition, a martingale adapted to the brownian filtration; and the brownian filtration is, by definition, the filtration generated by the wiener process.