Single jump filtrations and local martingales
We define a single jump filtration generated by a nonnegative random variable on a given probability space. We prove a simple characterization of local martingales on this filtered space. This result seems to be new even if the filtration is the smallest one with respect to which the generating random variable is a stopping time. As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.