We consider some stochastic models that have been proposed for the trajectories of moving objects, including Brownian motion. This leads to the development of a general approach for dealing with paths including the use of functional stochastic differential equations. We then present an empirical example based on the surface drifting movements of a small satellite-linked radio transmitter tag after it detached from a whale shark in the western Indian Ocean. The daily estimates of the tag’s locations were determined from transmissions received at irregular times by polar-orbiting satellites of the Argos Data Collection and Location Service system. An aspect of the empirical analysis is a study of how well the sea surface currents that are derived from remote sensing and sea surface models compare to the movements of the drifting tag. A second is to develop a predictive model using the past tag locations, the currents and the winds.