The past of a stopping point and stopping for two-parameter processes

Optional increasing paths passing through a given stopping point are studied. A characterization of the two extreme optional increasing paths is obtained. The past of a stopping point is defined, and a description of the largest stopping point smaller than two given stopping points is given. A stopping procedure is naturally associated with this notion of infimum. Stopped martingales and stopped filtrations are studied. "Local martingales" are defined and studied along horizontal and vertical lines. A nontrivial example of "local martingale" is given.