The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes

For a wide class of local martingales (M-t) there is a default function, which is not identically zero only when (M-t) is strictly local, i.e. not a true martingale. This 'default' in the martingale property allows us to characterize the integrability of functions of sup(s less than or equal to t) M-s in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales, and the default functions for Bessel processes and radial Ornstein-Uhlenbeck processes in relation to their first hitting and last exit times. Mathematics Subject Classtfication (1991): 60G44, 60J60, 60H15.