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

For a wide class of local martingales (Mt) there is a default function, which is not identically zero only when (Mt) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability of functions of sups≤tMs 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.